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Algebra Univers. (2023) 84:17 c 2023 The Author(s) Algebra Universalis https://doi.org/10.1007/s00012-023-00815-7 Topology of closure systems in algebraic lattices Niels Schwartz Abstract. Algebraic lattices are spectral spaces for the coarse lower topol- ogy. Closure systems in algebraic lattices are studied as subspaces. Con- nections between order theoretic properties of a closure system and topo- logical properties of the subspace are explored. A closure system is alge- braic if and only if it is a patch closed subset of the ambient algebraic lattice. Every subset X in an algebraic lattice P generates a closure sys- tem X . The closure system Y generated by the patch closure Y of P P X is thepatch closureof X .If X is contained in the set of nontrivial prime elements of P then X isaframe andisacoherentalgebraic frame if X is patch closed in P . Conversely, if the algebraic lattice P is coherent then its set of nontrivial prime elements is patch closed. Mathematics Subject Classification. 06A15, 06A06, 06B35. Keywords. Poset, Complete lattice, Algebraic lattice, Frame, Closure sys- tem, Closure operator, Spectral space, Specialization, Coarse lower topol- ogy, Scott topology, Patch topology. 1. Introduction Algebraic lattices, [4, Section I.4], [8, Section I-4], [9, p. 106, Definition 12], and closure systems, [5, section II.1], [4, Section I.5], [8, p. 26 ff], are classical topics in lattice theory and poset theory. Algebraic lattices are a class of complete lattices and abound in algebra, cf. [6, Examples 7.2.13], [8, Theorem I−4.16]. By [6, Theorem 7.2.8] every algebraic lattice is a spectral space for its coarse lower topology (see [6, p. 589] or 2.9 for the definition). Closure systems in algebraic lattices are studied as subspaces. Every topological space T carries a binary relation , which is called specialization and is defined by: x y if y ∈ {x},cf. [6, 1.1.3]. It is well known that the specialization relation is a partial order if and only if X is a T -space, Presented by W.W. McGovern. 0123456789().: V,-vol 17 Page 2 of 33 N. Schwartz Algebra Univers. and then specialization establishes connections between topological spaces and posets. Moreover, a T -topology on a poset is a lower topology, resp. an upper topology, if the specialization order is the given partial order, resp. the inverse partial order. Every poset P has at least one, frequently many, upper and lower topologies. Some particularly important examples are described in 2.9, e.g., the coarse lower topology, τ (P ), and the Scott topology, σ(P ) (which is an upper topology). Let P be a poset. See [8, Definition I−1.1] for the notion of compact elements in P . The set of compact elements is denoted by K(P ). In particular, if T is a topological space and O(T ) is the frame of open sets then K(O(T )) = K(T ) is the set of quasi-compact open subsets of T . For a bounded lattice P the bottom element is compact and the join of two compact elements is compact, but the top element and the meet of two compact elements need not be compact. If these are also compact then P is called coherent. A complete lattice P is algebraic if every element is the supremum of the compact elements below it. Thus an algebraic lattice has many compact elements, and K(P)is a decisive part of its structure. If P is an algebraic lattice then τ (P)isa spectral topology and σ(P ) is its inverse topology, [6, Theorem 7.2.8]. Recall that a topological space T is spectral if it is T and sober and K(T ) is a basis of open sets and a bounded sublattice of the frame of open sets, [6, Definition 1.1.5]. An important part of the structure of a spectral space is the patch topology (or constructible topology ), which is the Boolean topology generated by the quasi-compact open sets and their complements, [6, Section 1.3]. The coarse lower topology and the Scott topology are defined for arbitrary posets, but usually are not spectral topologies. Although the focus of the paper is on algebraic lattices, various results can and will be proved for closure systems in larger classes of posets. The first two sections are of a preparatory nature. They fix notation, recall terminology and exhibit various elementary facts about posets and clo- sure systems. Section 2 deals with posets and topologies on posets. Morphisms between posets are the monotonic maps. Order-theoretic properties of posets and poset maps are related to topological properties, several typical results are contained in Theorem 2.11. Closure operators and closure systems in posets are discussed in Section 3. Closure systems in a poset P correspond bijectively to closure operators. The closure operator P → P belonging to a closure sys- tem Q ⊆ P is denoted by η . Conversely, the image of a closure operator Q,P η : P → P is a closure system. The set of closure systems is denoted by C (P ). The algebraic closure systems are a particularly important subset of C (P ), which is denoted by A (P ). If P is a complete lattice then C (P)and A (P)are both closure systems in the power set P(P ). Thus, every subset X ⊆ P gener- ates a closure system, which is denoted by X . First topological properties of closure systems appear in Remark 3.6 and Proposition 3.7. For example, if P is a poset and Q is a subset then both P and Q carry their respective coarse lower topologies and Scott topologies. It is always true that τ (Q) ⊆ τ (P )| , and equality holds if Q is a closure system. The relation σ(Q) ⊆ σ(P )| holds Q Topology of closure systems Page 3 of 33 17 for all closure systems, and equality holds if and only if Q is an algebraic closure system. The analysis of closure systems in algebraic lattices starts with Section 4.If Q ⊆ P is a closure system in an algebraic lattice then it is always true that K(Q) ⊆ η (K(P )), Proposition 4.2. It is algebraic if and only if it is Q,P patch closed in P , if and only if K(Q)= η (K(P )), Theorem 4.5. Q,P Generating sets of closure systems enter the picture in Section 5.Let X be any subset in an algebraic lattice P and Y the patch closure of X. Then Y is the patch closure of X in P , and the compact elements of Y P P P are contained in X (but need not be compact in X ), Theorem 5.2. Sec- P P tion 6 is devoted to complete lattices and closure systems generated by sets of prime elements. An element p ∈ P is prime if a ∧ b ≤ p implies a ≤ p or b ≤ p,[8, Proposition I−3.12]. The set of prime elements is denoted by P(P ). The top element is always prime, the trivial prime element. The set of non- trivial prime elements is denoted by P(P ) and may be empty. The inclusion P(P ) ∩ Q ⊆ P(Q) holds for any closure system Q ⊆ P and may be proper. But if Q = P(Q) then Q is isomorphic to the frame O(P(Q) ), Theorem 6.13, and the equality P(P ) ∩ Q = P(Q) holds if and only if the closure operator η is a ∧-homomorphism, Corollary 6.9. The set of closure systems satisfy- Q,P ing these conditions is denoted by N (P ). The poset N (P ) (with inclusion) is isomorphic to the set of closed sets for the b-topology associated with τ (P ), Remark 6.15. Coherent algebraic lattices are particularly important in appli- cations. It is shown in Theorem 6.20 that P(P ) ⊆ P is patch closed if P is a coherent algebraic frame. On the other hand, if P is an algebraic lattice, X is patch closed in P and is contained in P(P ) then X is a coherent algebraic frame. In Section 7 results of the previous sections are specialized to frames. If P is a frame and a closure system Q belongs to N (P ) then the closure operator η is a nucleus of P.If P is a coherent algebraic frame then A (P ) ∩ N (P ), Q,P the set of algebraic closure systems in N (P ), is isomorphic (as a poset) to the set of patch closed subsets of P(P ) and is a closure system in P(P ), Theorem 7.1. The literature about posets, complete and algebraic lattices, frames, and closure systems is huge. We mention [4, 5, 8, 9, 11, 15] as general references. The terminology and notation for posets is mostly the same as in [6, Appendix]. Everything needed about spectral spaces can be found in [6]. 2. Topological spaces and posets To study connections between posets and topological spaces it is first necessary to fix the notation and terminology and note some elementary facts. In partic- ular we mention the b-topology associated with any topological space and its connections with irreducible sets, generic points and soberness, cf. 2.2 and 2.3. We recall the specialization order in T -spaces, cf. 2.8, as well as intrinsic topologies,cf. [14], on posets, most importantly the coarse lower topology,the Scott topology and the Lawson topology, 2.9. The morphisms between posets 17 Page 4 of 33 N. Schwartz Algebra Univers. are the monotonic maps, cf. 2.5. A continuous map between T -spaces is mono- tonic for the specialization order. Various order-theoretic properties of a poset map are related to continuity properties with respect to suitable upper and lower topologies, cf. Theorem 2.11 and its corollaries. We pay particular at- tention to left adjoint and right adjoint maps of poset maps. They are used to define closure systems and are decisive tools for their study. 2.1. Notation for topological spaces For a topological space T =(T, τ ) the frame of open sets is denoted by O(T)or O(τ ), and A(T)= A(τ ) is the set of closed sets. A continuous map f : S → T yields the inverse image maps O(f): O(S) →O(T)and A(f): A(S) →A(T ). 2.2. The b-topology associated with a topological space The locally closed sets in a topological space T =(T, τ ) are a basis of open sets for the b-topology, denoted by β = β(τ ), cf. [19, 2.2] or [6, 4.5.20]. We call β(T)=(T, β(τ )) the b-space of T.The b-topology has also been studied under the name Goldman topology or G-topology, see [3, section 4], in particular loc.cit., Proposition 4.2. If B is a basis of open sets for τ then the sets {x}∩ O with O ∈B and x ∈ T are a basis of open sets for β(τ ). The closure of A ⊆ T β β(τ ) for the b-topology is denoted by A = A ,and A is said to be very dense in a β-closed set B ⊆ T if A = B. A continuous map f : S → T is also continuous as a map of the associated b-spaces. In particular, if S is a subspace of T then β(S) ⊆ β(T ) is a subspace. 2.3. Irreducible sets, sober spaces, and sobrification A T -space T is sober if every nonempty closed and irreducible subset C is equal to {c} for a unique c ∈ C,the generic point of C.Every T -space is contained in a smallest sober space, its sobrification. The sobrification of T is denoted by Sob(T ), and Sob : T → Sob(T ) is the inclusion map. The sobrification was first studied in [10, Section 0.2.9] where it is presented as the set of nonempty closed and irreducible subsets of T with a suitable topology. Another presentation is given in [6, Section 11.2], which also contains a collec- tion of basic facts of the sobrification. In particular, the subspace T ⊆ Sob(T ) is very dense, [6, Corollary 11.2.4]. The paper [3] presents various results about the sobrification and connections with the b-topology. In particular, loc.cit., Theorem 4.16 and Theorem 4.20 show that for any sober space T (a) the sober subspaces are the b-closed subsets, and (b) for any subspace S the inclusion S → S is isomorphic to the sobrifica- tion. By [2], a continuous map f : S → T of T -spaces is an epimorphism in the category of T -spaces if and only if f (S) ⊆ T is very dense. Thus, Sob(S)is (up to isomorphism) the largest epimorphic extension of S and is contained in any sober space containing S. Topology of closure systems Page 5 of 33 17 2.4. Notation and terminology for posets Let P =(P, ≤) be a poset. Top and bottom elements in P , if they exist, are denoted by and ⊥ = ⊥ . If both exist then the poset is bounded. P P For a subset X ⊆ P we define X = X\{ } and X = X ∪{ }. (Nothing P P happens if does not exist.) Every a ∈ P generates a principal upset, a , and a ↓ ↑ ↑ ↓ ↓ principal downset, a .For S ⊆ P we set S = a and S = a .The a∈S a∈S ⇑ ↑ ⇓ ↓ sets of upper, resp. lower, bounds of S are S = a , resp. S = a . a∈S a∈S ↑Q ↑ ↑Q ↑ ⇑Q ⇑ For Q ⊆ P we define a = a ∩ Q, S = S ∩ Q, S = S ∩ Q, and so on. The set of all upsets, resp. principal upsets, are denoted by ⇑(P ), resp. ↑(P ). We call P a dcpo, resp. an fcpo,if D exists for every up-directed D ⊆ P , resp. D exists for every down-directed D ⊆ P,cf. [8,0−2.1]. Note that up-directed and down-directed sets are non-empty by definition. For the way-below relation on P and the definition of compact elements we refer to [8,I−1.1] or [6, 7.1.1]. If a ∈ P then a = a is the set of elements that are way below a and a = a = {x ∈ P | a x}. The set of compact elements is denoted by K(P ). Bottom elements, if they exist, are trivially compact. There need not be any other compact elements. Now assume that P is a bounded lattice. Then K(P ) is always a ∨-subsemilattice of P.Ifthe meet of two compact elements is compact (i.e., K(P ) ⊆ P is a sublattice) then P is arithmetic,cf. [8,I−4.7]. Moreover, P is coherent if K(P ) is a bounded sublattice, cf. [1, p. 2]. The way below relation and compact elements are used in [8, Definitions I−1.6 and I−4.2] to define continuous lattices (a complete lattice P such that a is up-directed for each a ∈ P and a = a )and algebraic lattices (see the introduction). 2.5. Poset morphisms A map ϕ : P → Q between posets is a poset morphism if it is monotonic. A poset map may have additional properties, which will always be announced explicitly. • ϕ is a dcpo-homomorphism if ϕ(D) exists and is equal to ϕ( D) Q P for all up-directed D ⊆ P such that D exists. The notion of fcpo- homomorphisms is defined similarly. • ϕ is a ∧-homomorphism if ϕ(a) ∧ ϕ(b) exists and is equal to ϕ(a ∧ b) Q P for all a, b ∈ P such that a ∧ b exists. Homomorphisms for , ∨ and are defined accordingly. Clearly, a -map, resp. a -map, is also a ∧-map and an fcpo-map, resp. a ∨-map and a dcpo-map. • ϕ is a poset-embedding if x ≤ y is equivalent to ϕ(x) ≤ ϕ(y)for x, y ∈ P , resp. a ∧-embedding if it is a poset-embedding, a ∧-morphism and x ∧ y exists if ϕ(x) ∧ ϕ(y) exists. Embeddings for , ∨, , as well as dcpo- embeddings and fcpo-embeddings are defined similarly. • ϕ : P → Q is coherent if ϕ(K(P )) ⊆ K(Q), cf. [1, p. 2]. If ϕ is a poset map then the inverse image map P(ϕ): P(Q) → P(P)of the power sets restricts to a poset map ⇑(ϕ): ⇑(Q) →⇑(P ). The inclusion P(ϕ)(↑(Q)) ⊆↑(P ) need not hold, but if it holds then ↑(ϕ): ↑(Q) →↑(P ) denotes the restriction of P(ϕ). 17 Page 6 of 33 N. Schwartz Algebra Univers. 2.6. Adjoint pairs of poset morphisms See [8,p.22ff]and [7, p. 155 ff] for the following notions and facts. Consider poset morphisms ϕ : P → Q and ψ : Q → P . We say that (ϕ, ψ)isan adjoint pair with ψ right adjoint to ϕ and ϕ left adjoint to ψ if the equivalence ϕ(a) ≤ −1 ↓ ↓ b ⇔ a ≤ ψ(b) holds for all a ∈ P and b ∈ Q, if and only if ϕ (b )= ψ(b) −1 ↑ ↑ for all b ∈ Q, if and only if ψ (a )= ϕ(a) or all a ∈ P . Each member of the pair (ϕ, ψ) determines the other one uniquely. The map ϕ has a right adjoint −1 ↓ ϕ if and only if for all b ∈ Q the set ϕ (b ) has a largest element (which then is equal to ϕ (b)). Similarly, ψ has a left adjoint ψ if and only if for all −1 ↑ ∗ a ∈ P the set ψ (a ) has a smallest element (which then is equal to ψ (a)), if and only if the map ↑(ψ) is defined, cf. 2.5. Let (ϕ, ψ) be an adjoint pair. (a) If a ∈ P and b ∈ Q then a ≤ ψ ◦ ϕ(a)and ϕ ◦ ψ(b) ≤ b. (b) ϕ ◦ ψ ◦ ϕ = ϕ and ψ ◦ ϕ ◦ ψ = ψ,[7, p. 159, Lemma 7.26]. (c) ψ is injective if and only if ϕ ◦ ψ =id , if and only if ϕ is surjective, whereas ψ is surjective if and only if ψ ◦ ϕ =id , if and only if ϕ is injec- tive, [8, Proposition 0−3.7]. If ψ is injective then it is a poset-embedding. For, x, y ∈ Q and ψ(x) ≤ ψ(y) implies x = ϕ ◦ ψ(x) ≤ ϕ ◦ ψ(y)= y.The same holds for ϕ. If P is complete then ϕ : P → Q has a right adjoint, resp. left adjoint, if and only if it is a -morphism, resp. a -morphism, [8, Corollary 0−3.5]. Proposition 2.7. Let (ϕ : P → Q, ψ : Q → P ) be an adjoint pair. (a) ψ is a -morphism and ϕ is a -morphism. (b) If ψ is injective then it is a -embedding, hence a ∧-embedding and an fcpo-embedding. If ϕ is injective then it is a -embedding, hence a ∨- embedding and a dcpo-embedding. (c) If ψ is injective and a -homomorphism, resp. a ∨-homomorphism, resp. a dcpo-morphism, then it is a -embedding, resp. a ∨-embedding, resp. a dcpo-embedding. If ϕ is injective and a -homomorphism, resp. a ∧- homomorphism, resp. an fcpo-morphism, then it is a -embedding, resp. a ∧-embedding, resp. an fcpo-embedding. Proof. It suffices to prove the assertions about ψ; the proofs for ϕ are similar. (a). Pick S ⊆ Q such that S exists. Then ψ( S) ≤ ψ(s) for all Q Q s ∈ S.Pick a ∈ P with a ≤ ψ(s) for all s ∈ S. Then ϕ(a) ≤ s for all s ∈ S, hence ϕ(a) ≤ S. It follows that a ≤ ψ( S), i.e., ψ( S)= ψ(S). Q Q Q P (b). Note that ψ is a poset-embedding by 2.6(c) and is a -morphism by (a). Assume S ⊆ Q and a = ψ(S) exists. We show that ϕ(a)= S. P Q For all s ∈ S we have ϕ(a) ≤ s.Pick b ∈ Q with b ≤ s for all s ∈ S. Then ψ(b) ≤ ψ(s) for all s, hence ψ(b) ≤ a,and 2.6(c) implies b = ϕ ◦ ψ(b) ≤ ϕ(a). It follows that S exists and is equal to ϕ(a). Clearly, ψ is a ∧-embedding and an fcpo-embedding. (c). Again, ψ is a poset-embedding by 2.6(c) and is a -map by hy- pothesis. Now suppose S ⊆ Q and a = ψ(S) exists. By 2.6(a) we have ψ(s) ≤ a ≤ ψ ◦ ϕ(a) for all s ∈ S. Since ψ is a poset-embedding it follows that Topology of closure systems Page 7 of 33 17 s ≤ ϕ(a) for all s ∈ S.Now let b ∈ Q be an upper bound for S. Then ψ(b)is an upper bound for ψ(S), and we see that a ≤ ψ(b). It follows that ϕ(a) ≤ b. If ψ is a ∨-map, resp. a dcpo-map, then use subsets S ⊆ Q with |S| =2, resp. up-directed sets S ⊆ Q. 2.8. Specialization in topological spaces See the Introduction for the definition of the specialization relation of a topo- logical space. Every T -space is considered as a poset via specialization, and {x} = x . Continuous maps between T -spaces are monotonic for specializa- tion. Consider a T -space T and a nonempty irreducible subset C.If C has the generic point c,cf. 2.3, then C = c and C exists and is equal to c,[6, Proposition 4.2.1(ii)]. Every down-directed set in T is nonempty and irreducible, [6, Proposition 4.2.1(i)], but irreducible sets need not be down- directed, [6, Example 4.2.3]. We say that T is (a) sober for down-directed sets if C has a generic point whenever C ⊆ T is down-directed; (b) sober for irreducible sets with infimum if C has a generic point for all nonempty irreducible C such that C exists; (c) sober for down-directed sets with infimum if C has a generic point for all down-directed sets C such that C exists. Note the following simple facts. (d) C ⊆ T has an infimum if and only if C has an infimum. Then the infima are equal. (e) Let f : S → T be continuous, C ⊆ S a nonempty irreducible set and as- sume that C has a generic point, namely C. Then f (C) is nonempty irreducible and f (C) = f ( C) , hence f ( C)= f (C). S S T (f) Let S ⊆ T a subspace, C ⊆ S nonempty and irreducible. Then C has a generic point if and only if C has a generic point and C ∈ S. (g) Let T be sober and S ⊆ T a subspace. Then S is sober if and only if C ∈ S for all nonempty irreducible C ⊆ S. 2.9. Lower and upper topologies on posets A topology on the poset P is said to be intrinsic,cf. [14], if it can be defined in terms of the partial order. A T -topology on P is a lower topology,oran upper topology if its specialization order is ≤, resp. the inverse partial order ≤ , inv cf. [6, p. 589]. Every open set of a lower topology is a downset for ≤,every open set of an upper topology is an upset. If (T, τ)isa T -space then τ is a lower topology for .The coarse lower topology τ (P ) and the coarse upper topology τ (P ) (for the definition see [6, p. 589] ) are intrinsic topologies. The set ↑(P)(see 2.5 for the notation) is called the canonical subbasis of closed sets for the coarse lower topology. If τ is any lower topology then ↑(P ) is the set of nonempty closed and irreducible sets with generic point. We need two more intrinsic topologies: 17 Page 8 of 33 N. Schwartz Algebra Univers. (a) A set U ⊆ P is open for the lower Scott topology σ (P)if U = U and U ∩ D = ∅ for each down-directed D ⊆ P such that D exists and belongs to U . (b) A set U ⊆ P is open for the Scott topology σ(P)if U = U and U ∩ D = ∅ for each up-directed set D ⊆ P such that D exists and belongs to U . (This definition is slightly more general than the usual one, where it is assumed that P is a dcpo, [8, Definition II−1.3], [14, p. 82], [6, Definition 7.1.6].) In particular, a principal upset a is Scott open if and only if a ∈ K(P ). For the case of a dcpo basic properties of τ (P ), resp. σ(P ), are presented in [6, 7.1.4], resp. [8, section II-1] or [6, 7.1.6 and 7.1.8]. The coarse upper topology (resp. the lower Scott topology) is the coarse lower topology (resp. the Scott topology) for the inverse partial order ≤ .Thuspropertiesof τ (P ) inv and σ (P ) follow from properties of τ (P)and σ(P ), and vice versa. Note the following useful fact. (c) A lower topology is coarser than the lower Scott topology if and only if it is sober for down-directed sets with infimum. Let τ be an upper topology on P . Then item (c) says that τ ⊆ σ(P)ifand only if D has a generic point for every up-directed set D with supremum. This equivalence strengthens [6, 7.1.8(x)], where it is shown that every spectral upper topology on P is coarser than the Scott topology. The join of τ (P)and σ(P ) in the lattice of topologies on P is the Lawson topology,[8, Definition III−1.5], which is denoted by λ(P ). A detailed discus- sion of the Lawson topology is contained in [8, Chapter III], always under the assumption that P is a dcpo. However, various results and arguments are true with the more general definition used here. The sets U \F with U ∈O(σ(P )) and F ⊆ P finite are a basis of open sets for the Lawson topology, [8,p. 211 f.]. The Lawson topology is coarser than both b-topologies β(τ (P )) and β(σ(P )). An upset is Lawson open if and only if it is Scott open, [8, Proposition III−1.6(a)]. Let ϕ : P → Q be a poset map. If P and Q are both equipped with their coarse lower topologies, resp. their Scott topologies, and so on, and ϕ is con- tinuous then we say that ϕ is coarse lower continuous, resp. Scott continuous, and so on. 2.10. Algebraic lattices as spectral spaces An algebraic lattice P is a spectral space for the coarse lower topology, and the Scott topology is the inverse topology, [6, Theorem 7.2.8]. A subset U ⊆ P is quasi-compact open if and only if U = P \F with F ⊆ K(P ) finite, loc. cit..Thusthe sets a with a ∈ K(P ) are a subbasis of closed sets for τ (P ). Usually this is a proper subset of the canonical subbasis ↑(P ), cf. 2.9. The patch topology of the spectral space P is the join of the spectral topology and the inverse topology, hence is equal to λ(P ). For any subset con X ⊆ P the patch closure of X is denoted by X .The basic constructible ↑ ↑ sets F \ G , with F, G ⊆ K(P ) finite, are a basis of open sets for the patch ↑ ↑ ↑ ↑ ↑ ↑ topology. Since F \G = a \G the sets a \ G (with a ∈ K(P)and a∈F Topology of closure systems Page 9 of 33 17 ↑ ↑ G ⊆ K(P ) finite) are a basis as well, and the sets a \b (where a, b ∈ K(P )) are a subbasis. Note that K(P ) is dense in P for the patch topology since a ↑ ↑ ↑ ↑ basic open set a \ G is nonempty if and only if a ∈ a \G . Theorem 2.11. Let ϕ : P → Q be a poset map. Then: (a) Let ϕ :(P, τ ) → (Q, τ ) be continuous, where τ and τ are lower P Q P Q topologies on P and Q.If τ ⊆ σ (P ) then ϕ is an fcpo map. (b) ϕ is an fcpo map if and only if it is lower Scott continuous. (c) If ϕ has a left adjoint then it is coarse lower continuous. (d) Let ϕ :(P, τ ) → (Q, τ ) be continuous, where τ and τ be upper P Q P Q topologies on P and Q.If τ ⊆ σ(P ) then ϕ is a dcpo map. (e) (cf. [8, Proposition II−2.1]) ϕ is adcpomap if andonlyifitisScott continuous. (f) If ϕ has a right adjoint then it is coarse upper continuous. (g) If ϕ is Lawson continuous then it is Scott continuous (cf. [8, Theorem III−1.8]) and lower Scott continuous. Proof. (a). If D ⊆ P is down-directed and D exists, then D =( D) , P P cf. 2.9(c), and ϕ( D)= ϕ(D) holds by 2.8(e). P Q (b). Assume that ϕ is an fcpo-map and let V ⊆ Q be σ (Q)-open. It is −1 claimed that the downset U = ϕ (V ) ⊆ P is σ (P )-open. If D ⊆ P is down- directed such that D exists and belongs to U then ϕ(D)= ϕ( D) ∈ P Q P V , hence ϕ(D) ∩ V = ∅, i.e., D ∩ U = ∅. – The converse holds by (a). (c) follows from 2.6, and (d), (e), (f) are proved exactly as (a), (b), (c), mutatis mutandis (or one applies (a), b), (c) to the inversely ordered sets P inv and Q ). inv −1 (g). For Scott continuity, let V ⊆ Q be Scott open. Then U = ϕ (V ) is an upset in P and is Lawson open, hence is Scott open, cf. 2.9.Toprove lower Scott continuity we show that ϕ is an fcpo-map, cf. item (b). So, assume D ⊆ P is down-directed and D exists. It is claimed that ϕ( D)isthe P P with b ≤ ϕ( D). infimum of ϕ(D). If this is false there is some b ∈ ϕ(D) ↑Q −1 The set V = Q\b is open for τ (Q), hence is Lawson open, and U = ϕ (V ) is Lawson open in P,say U = U \F (with U Scott open and F ⊆ P i i i i∈I i ↑Q finite). Since ϕ(D) ⊆ b and b ≤ ϕ( D) we see that D ∈ U and P P D ∩ U = ∅.Pick i ∈ I such that D ∈ U \F and let F = {x ,...,x }. i i 1 l For each j =1,...,l there is some d ∈ D ∩ P \x . For, otherwise x ∈ D j j and x ≤ D, which is false. If d ∈ D is a lower bound for {d ,...,d } then j 1 l ↑ ↑ ↑ d ∈ P \ F .Thus U = U and D ∈ U imply d ∈ D ∩ (U \F ) ⊆ D ∩ U,a i i i i i P i contradiction. Remark 2.12. Let X and Y be spectral spaces and f : X → Y a spectral map. Then f is monotonic for the specialization order and is both a dcpo map and an fcpo map, cf. 2.8(e). The spectral topologies of X and Y are coarser than the lower Scott topologies and the inverse topologies are coarser then the Scott topologies, [6, 7.1.8(x)]. It follows from Theorem 2.11(b) and (e) that f is continuous both for the lower Scott topology and the Scott topology. 17 Page 10 of 33 N. Schwartz Algebra Univers. Corollary 2.13. Let ϕ : P → Q be a monotonic map of complete lattices. (a) The following conditions are equivalent. (i) ϕ preserves all infima. (ii) ϕ preserves finite infima and is coarse lower continuous. (iii) ϕ preserves finite infima and is lower Scott continuous. (b) The following conditions are equivalent. (i) ϕ preserves all suprema. (ii) ϕ preserves finite suprema and is coarse upper continuous. (iii) ϕ preserves finite suprema and is Scott continuous. (c) The following conditions are equivalent. (i) ϕ preserves all infima and is a dcpo map. (ii) ϕ preserves finite infima and is both coarse lower continuous and Scott continuous. (iii) ϕ preserves finite infima and is continuous for the Lawson topology. If the equivalent conditions hold then the left adjoint map ϕ exists and is coherent. Proof. (a). (i)⇒(ii). The first statement in (ii) is trivial. Coarse lower conti- nuity follows from the existence of a left adjoint, cf. 2.6 and Theorem 2.11(c). The implications (ii)⇒(iii) and (iii)⇒(i) follow from Theorem 2.11(a) and (b). (b). Apply item (a) to ϕ : P → Q . inv inv (c). (i)⇒(ii) follows from item (a) and Theorem 2.11(e). (ii)⇒(iii) is triv- ial. Finally, item (a) and Theorem 2.11(e) and (g) yield (iii)⇒(i). If (i)–(iii) hold then ϕ exists by 2.6 and condition (i). An element x in a poset is com- ↑ ∗ ↑ −1 ↑ pact if and only if x is Scott open, 2.9(b). Thus, ϕ (b) = ϕ (b ) is Scott open if b ∈ K(Q), and then ϕ (b) is compact. Corollary 2.14. Let ϕ : P → Q be a bounded lattice homomorphism of algebraic lattices. (a) ϕ is spectral if and only if it preserves all infima and all suprema. (b) If ϕ exists then it is spectral if and only if ϕ is coherent. Proof. (a). Use Corollary 2.13. −1 ↑ ↑ (b). If a ∈ P then ϕ (a )= ϕ(a) ⊆ Q,see 2.6. Assume ϕ is spectral and a ∈ K(P ). Then ϕ(a) ⊆ Q is closed and constructible, hence ϕ(a) ∈ K(Q). ↑ ↑ Now let ϕ be coherent. If a ∈ K(P ) then ϕ(a) ∈ K(Q), hence a and ϕ(a) are −1 closed and constructible. Thus ϕ (C) is closed and constructible if C ⊆ P is closed and constructible. 2.15. Products of posets Let P and Q be bounded posets. By [8, Lemma III−1.3] the coarse lower topology on P × Q is the product of the coarse lower topologies on P and Q. The projection π : P × Q → P (similarly the projection π ) has both P Q right and left adjoints and (π ) (a)=(a, )and π (a)=(a, ⊥ ). For P ∗ Q Q x ∈ P × Q let x = π (x)and x = π (x). A subset D ⊆ P × Q is up- P P Q Q directed if and only if D = {d | d ∈ D} and D = {d | d ∈ D} are P P Q Q up-directed, and x = D if and only if x = D , x = D . P P Q Q P ×Q P Q Topology of closure systems Page 11 of 33 17 ∗ ∗ Thus K(P × Q)= K(P ) × K(Q)and π , π , π , π are all coherent. It P Q P Q follows that P × Q is an algebraic lattice if so are P and Q. If Q = P then the diagonal map Δ : P → P × P, x → (x, x) is both a -homomorphism and a -homomorphism and is coherent. (a) The right adjoint Δ exists if and only if P is a ∧-semilattice. If this is the case then Δ (x, y)= x ∧ y and Δ is coarse lower continuous (also see ∗ ∗ [8, Lemma III−1.4]), hence is an fcpo-map and is lower Scott continuous, Theorem 2.11(a), (b), (c). (b) The left adjoint Δ exists if and only if P is a ∨-semilattice. Then ∗ ∗ Δ (x, y)= x ∨ y and Δ is continuous for the coarse upper topology, hence is a dcpo map and is Scott continuous, Theorem 2.11(d), (e), (f). Moreover, Δ is coherent. The Scott topology of products is more complicated. Usually σ(P × Q) is finer than the product of σ(P)and σ(Q), [8, p. 197]. Corollary 2.16. Let P be an algebraic lattice. Then Δ and ∧ =Δ are spectral maps. Proposition 2.17. Let P be a frame. Then ∧ : P × P → P is Scott continuous, hence also Lawson continuous. Proof. Scott continuity is equivalent to Lawson continuity since ∧ is coarse lower continuous, 2.15(a), Theorem 2.11(g). For Scott continuity pick a Scott −1 open set U ⊆ P and let D ⊆ P × P be up-directed with D ∈∧ (U ). P ×P Define D and D to be the sets of first and second components of elements 1 2 of D and let C = {x ∧ y | x, y ∈ D}⊆ P , which is up-directed as well. Then 1 2 D =( D , D )and ∧( D)= D ∧ D = C (as P 1 2 1 2 P ×P P P P ×P P P P is a frame). Thus, C ∈ U implies the existence of x, y ∈ D with x ∧y ∈ U . 1 2 ↑ ↑ For any z ∈ x ∩ y ∩ D we have x ∧ y ≤ z ∧ z = ∧(z), and it follows that 1 2 1 2 −1 z ∈ D ∩∧ (U ). 3. Closure operators and closure systems In this section we fix the notation and terminology concerning closure systems and closure operators, exhibit some examples and present first topological properties of closure systems, cf. Remark 3.6 and Proposition 3.7. Notation and Terminology 3.1. A poset map η : P → P is a closure operator if it is idempotent and inflationary (i.e., a ≤ η(a)), resp. a kernel operator if it is idempotent and deflationary (i.e., η(a) ≤ a), [8, Definition 0−3.8]. It suffices to discuss closure operators since a poset map η : P → P is a closure operator if and only if η viewed as a map P → P (the inversely ordered poset) is inv inv a kernel operator. Let Q ⊆ P and ι = ι : Q → P the inclusion map. Then Q is a closure P,Q ↑Q system if ι has a left adjoint, equivalently if, for all a ∈ P , the set a has a smallest element, cf. 2.6. The left adjoint ι : P → Q is denoted by ϑ = ϑ Q,P and is called the closure map. The composition η = ι ◦ ϑ is a closure Q,P P,Q Q,P 17 Page 12 of 33 N. Schwartz Algebra Univers. operator. Conversely, if η : P → P is a closure operator then Q = η(P)is a closure system with η = η ,and ϑ is the corestriction of η. The cor- Q,P Q,P respondence between closure systems and closure operators of P is bijective. ↑Q ↑Q ↑Q ↑Q Clearly, η(a) = a and η(a)= a = a where a ∈ P . The inclu- P Q sion ι is a -embedding, Proposition 2.7(b), and is coarse lower continuous, Theorem 2.11(c). The closure map is a -map, hence a dcpo map, Proposition 2.7(a), hence is Scott continuous, Theorem 2.11(e). A closure system Q ⊆ P , and the corresponding closure operator, are algebraic,or inductive,if ι is P,Q a dcpo map (equivalently a dcpo embedding, Proposition 2.7(b)). The set of all closure systems, resp. algebraic closure systems, is denoted by C (P ), resp. A (P ). ↑X Consider a subset X ⊆ P such that a exists for all a ∈ P . Then ↑X η : P → P, a → a is a closure operator, X := η(P ) is the smallest closure system containing X and is called the closure system generated by X. In arbitrary posets not every subset generates a closure system. Here are two examples. First, let P = {a, b} with a, b incomparable, X = {a}. Then ↑X ↑X b = ∅, hence b does not exist since there is no top element in P.For the second example, let N be the inversely ordered set of natural numbers; its ∗ ∗ ∗ elements are denoted by 0 , 1 ,....Let P = N ∪ N with N naturally ordered ∗ ∗ ↑X ∗ and k ≤ l for all k, l ∈ N.If X = N then k = N for all k ∈ N, which does not have an infimum in P . We say that X is dense in P if X exists and is equal to P.If P is a complete lattice and Q ⊆ P then one shows easily that Q is a closure system if and only if S ∈ Q for all S ⊆ Q. In particular, Q is a complete lattice as well. It follows that C (P)and A (P ) are closure systems (not necessarily algebraic) in the complete lattice P(P ). If X ⊆ P then η (X)= X , and its elements are the infima of subsets C (P ),P(P ) S ⊆ X. Example 3.2. Let ψ : P → Q be the left adjoint of a poset map ψ : Q → P.It follows from 2.6 that ψ ◦ ψ : P → P is a closure operator with corresponding closure system ψ(Q), and ψ ◦ ψ is a kernel operator. Let C ⊆ P and D ⊆ Q be closure systems with inclusion maps ι , ι and closure maps ϑ , ϑ . Assume C D C D that ψ(D) ⊆ C and let ϕ : D → C be the restriction of ψ. Then ϕ has a left ∗ ∗ adjoint and ϕ ◦ ϑ = ϑ ◦ ψ .Moreover, ψ(D) is a closure system in P,and C D ψ(D)= ψ(X) if D = X . P Q Left adjoint maps are -homomorphisms, Proposition 2.7(a). Now as- sume that ψ is a dcpo map and D is an algebraic closure system, i.e., ι is a dcpo map. Then ϕ = ϑ ◦ ι ◦ ϕ = ϑ ◦ ψ ◦ ι is a dcpo map. Moreover, C C C D ψ(D) is also algebraic in P . To show this one may assume C = ψ(D), i.e., ϕ ∗ ∗ is surjective, and then ι = ι ◦ ϕ ◦ ϕ = ψ ◦ ι ◦ ϕ is a dcpo map, 2.6(c). C C D Example 3.3. If P is a poset then the set ↑(P ) generates a closure system A(P, ↑(P )) ⊆ P(P ), the set of ↑(P )-closed sets. The corresponding closure ↑(P ) ↑P map sends A ∈ P(P)toits ↑(P )-closure A = c . The top and c∈A Topology of closure systems Page 13 of 33 17 bottom elements of A(P, ↑(P )) are P , resp. P \ P . The complements of ↑(P )- closed sets in P are the ↑(P )-open sets, and the set of these is denoted by O(P, ↑(P )). Consider a poset map ψ : Q → P with left adjoint. The inverse image map P(ψ): P(P ) → P(Q) restricts to ↑(ψ): ↑(P ) →↑(Q), 2.6, hence yields the maps A(ψ, ↑): A(P, ↑(P )) →A(Q, ↑(Q)), O(ψ, ↑): O(P, ↑(P )) →O(Q, ↑(Q)). ∗ ∗ The left adjoints P(ψ) and A(ψ, ↑) , as well as the right adjoint O(ψ, ↑) ↑(P ) ∗ ∗ exist and P(ψ) (B)= {A ∈ P(P ) | ψ(B) ⊆ A}, A(ψ, ↑) (B)= ψ(B) , ↑(P ) O(ψ, ↑) (O)= Q\ψ(Q\O) . Let X ⊆ P be a subset with inclusion map e : X → P . We define ↑X ↑(X, P)= P(e)(↑(P )) = {a | a ∈ P}⊆ P(X) and call A(X, ↑(X, P )) = ↑(X, P ) the set of ↑(X, P )-closed subsets of P(X) X. The complements in X are the set O(X, ↑(X, P )) of ↑(X, P )-open subsets. The inclusion e yields the surjective maps A(e, ↑): A(P, ↑(P )) →A(X, ↑(X, P )), O(e, ↑): O(P, ↑(P )) →O(X, ↑(X, P )). The adjoints A(e, ↑) and O(e, ↑) exist and are both injective. The map γ : ∗ X,P ↑X P →O(X, ↑(X, P )),a → X\a is a poset morphism. The following lemmas record elementary facts about closure systems, generating sets and the maps γ (without proof). X,P Lemma 3.4. Let P be a poset, S, X ⊆ P and assume that Q := X exists. Then: ↑X ↑Q (a) a = a for all a ∈ P . P P ↑X ↑X (b) Q = {a ∈ P |∀b ∈ P : b ≤ a ⇒ a \ b = ∅}. ↑X ↑X (c) If S exists then ( S) = s . P P s∈S ↑(P ) ↑(P ) (d) S exists if and only if S ∈↑(P ), and then S =( S) . P P (e) P is complete if and only if A(P, ↑(P )) = ↑(P ). Lemma 3.5. Let P be a poset, X ⊆ P . (a) The poset map γ is an embedding if and only if X is dense in P . X,P (b) If P is complete then γ is surjective. The converse holds if X is dense X,P in P . Remark 3.6. Let P be a poset and Q ⊆ P a subset. Recall from [6, 7.1.4(ii)] that the inclusion τ (Q) ⊆ τ (P )| is always true and may be proper. Now assume that Q is a closure system in P with inclusion map ι, closure map ϑ and closure operator η. Since ι is coarse lower continuous (Theorem 2.11(c)) it follows that τ (Q)= τ (P )| . The closure map ϑ is a dcpo map and is Scott continuous, Theorem 2.11(d), (e), (f). (a) η is a dcpo map if ι is a dcpo map. The converse is true if P is a dcpo. 17 Page 14 of 33 N. Schwartz Algebra Univers. (b) η is a ∧-homomorphism if and only if ϑ is a ∧-homomorphism. Thus, the closure system is algebraic (by definition) if and only if ι is a dcpo map, if and only if ι is Scott continuous, and then η is Scott continuous (since the left adjoint ϑ = ι is a dcpo map, Proposition 2.7(a)). If P is a dcpo then Scott continuity of η implies that ι is Scott continuous, Theorem 2.11(e). Proposition 3.7. Consider a poset P with a closure system Q ⊆ P.Then: (a) Q is closed for β = β(τ (P )). (b) σ(Q) ⊆ σ(P )| . (c) σ(P )| = σ(Q), i.e., ι is Scott continuous, if and only if ι is Lawson continuous, if and only if Q is algebraic in P . (d) If Q is quasi-compact for λ(P ) then Q is an algebraic closure system. (e) If Q is a complete lattice and is an algebraic closure system then Q is quasi-compact for λ(P ). (f) If P is a complete lattice then Q is an algebraic closure system if and only if Q ⊆ P is quasi-compact for the Lawson topology. ↑P ↑P Proof. (a). If a ∈ P \Q then a<η(a), hence a belongs to a \ η(a) ,which is β-open and is disjoint from Q. ↓P (b). Let B be σ(Q)-closed in Q and set A = B .As B = A ∩ Q it suffices to show that A is σ(P )-closed. So, pick D ⊆ A up-directed such that D exists. For d ∈ D there exists b ∈ B with d ≤ b.Thus ϑ(d)= η(d) ≤ η(b)= b and ϑ(D) ⊆ B is an up-directed set in a Scott closed set. As ϑ is a dcpo-map we have D ≤ ϑ( D)= ϑ(D) ∈ B, which implies D ∈ A. P P Q P (c). The closure system is algebraic if and only if ι is a dcpo map, if and only if ι is Scott continuous, if and only if σ(P )| = σ(Q) (by (b)). Scott continuity of ι is equivalent to Lawson continuity (note that ι is coarse lower continuous and use Theorem 2.11(g)). (d). Let D ⊆ Q be up-directed such that D exists. We have to show that D exists and is equal to D. If this is false then there is some P Q ↓P ↓P a ∈ P with D ⊆ a , but D ≤ a. The set a is closed for σ(P ), hence ↓Q for λ(P ). Thus a is closed in Q for λ(P )| and is quasi-compact (as Q is ↑P ↓Q quasi-compact). The sets P \d with d ∈ D are open for λ(P ) and cover a . ↓Q ↑P (For, if x ∈ a ∩ d then D ≤ x ≤ a, a contradiction.) The cover d∈D Q k ↑P ↓Q ↑P ↓Q a ⊆ P \d has a finite subcover, say a ⊆ P \d .As D is d∈D i=1 i ↓Q ↑P up-directed there is some d ∈ D with d ,...,d ≤ d. Hence a ⊆ P \d and 1 k ↓Q ↑P D ⊆ a implies d ∈ P \ d , a contradiction. (e). Item (c) and Remark 3.6 show that λ(P )| = λ(Q). Thus it suffices to note that (Q, λ(Q)) is quasi-compact by [8, Theorem III−1.9]. (f) follows from (d) and (e). Proposition 3.8. Let P be a poset, C ⊆ P asubsetand τ a lower topology. (a) C is irreducible for τ (P ) if and only if, for all finite F ⊆ P with C ⊆ F , there is some x ∈ F with C ⊆ x . τ (P ) ↑(P ) (b) If C = ∅ is irreducible for τ (P ) then C = C (for the notation see Example 3.3),hence τ (P ) is sober for irreducible sets with infimum. In particular, τ (P ) is sober if P is complete. Topology of closure systems Page 15 of 33 17 Proof. (a). The sets F (where F ⊆ P is finite) are a basis of closed sets for ↑ ↑ ↑ τ (P ). If C is irreducible and C ⊆ F = x then C ⊆ x for some x ∈ F . x∈F Conversely, assume that C ⊆ x for some x ∈ F whenever F ⊆ P is finite and ↑ ↑ ↑ ↑ C ⊆ F . Pick finite subsets G, H ⊆ P with C ⊆ G ∪ H =(G ∪ H) . Then ↑ ↑ ↑ C ⊆ x for some x ∈ G ∪ H, hence C ⊆ G or C ⊆ H according as x ∈ G or x ∈ H,and C is irreducible. τ (P ) ↑(P ) (b). The equality C = y = C follows from (a). If C y∈C P exists then the set is equal to ( C) , Lemma 3.4(d). Corollary 3.9. Let P be a complete lattice. A subset Q is a closure system if andonlyifitisclosedin P under finite infima and is sober for τ (P )| . Proof. Assume Q is a closure system. Then it is closed in P under all infima. Proposition 3.7(a) shows that Q is β(τ (P ))-closed in P.As τ (P ) is sober, Proposition 3.8(b), it follows that Q is a sober subspace of P,see 2.3(a). Conversely, assume Q is sober for τ (P )| and is closed in P under finite infima. It is claimed that S ∈ Q for any subset S ⊆ Q.If S = ∅ then = S ∈ Q.For S = ∅ let T ⊆ Q be the set of all finite infima of elements of S. Then T is down-directed, hence is irreducible for any lower topology, [6, ⇓ ⇓ Proposition 4.2.1(i)], and S = T .As Q is sober it follows from 2.8(g) that ⇓ ⇓ S = S = T = T ∈ Q. P P P P 4. Algebraic closure systems in algebraic lattices Every algebraic lattice is a spectral space, 2.10. The main result in this section is Theorem 4.5, which contains several topological conditions characterizing algebraic closure systems in algebraic lattices. Proposition 4.1. Let P be a continuous lattice and Q ⊆ P an algebraic closure system. Then Q is a continuous lattice as well. Proof. By [8, Proposition I−1.5(ii)] we have to show that for each b ∈ Q there is an up-directed set D ⊆ b with D = b. By hypothesis, the set b is Q Q P up-directed and b = b .Thus, ϑ(b ) is up-directed and, since ϑ is a dcpo- P P P map, b = ϑ(b)= ϑ( b )= ϑ(b ). We show that ϑ(c) b for each P P Q P c ∈ b :Let D ⊆ Q be up-directed with b ≤ D = D. Then c b Q P implies c ≤ d for some d ∈ D, hence ϑ(c) ≤ ϑ(d)= d. Proposition 4.2. Let P be an algebraic lattice and Q ⊆ P a closure system. Then K(Q) ⊆ ϑ(K(P )). ↓K(P ) Proof. Pick a ∈ K(Q) and write a = a . Then ↓K(P ) ↓K(P ) a = ϑ(a)= ϑ a = ϑ(c) | c ∈ a P Q since ϑ is a dcpo map. As the supremum is up-directed and a ∈ K(Q) there is ↓K(P ) some c ∈ a with a ≤ ϑ(c) ≤ ϑ(a)= a. 17 Page 16 of 33 N. Schwartz Algebra Univers. Definition 4.3. Let P be an algebraic lattice and Q a closure system. An ele- ↓K(P ) ↓P ment a ∈ P is absorbing for Q,orfor η,if η(a ) ⊆ a . (This generalizes the notion of absorbing elements in an algebraic frame with respect to a nucleus as defined in [13, Definition and Remarks 4.1]). Proposition 4.4. Let Q ⊆ P be a closure system in an algebraic lattice and A the set of absorbing elements for Q.Then A is the patch closure of Q. Proof. The inclusion Q ⊆ A is obvious. First consider any a ∈ P \A. There ↓K(P ) ↑ ↑ is some c ∈ a with η(c) ≤ a. The set c \η(c) is patch open (by 2.10), con contains a and is disjoint from Q (by 3.1). It follows that Q ⊆ A. For the reverse inclusion pick a ∈ A and let C be a patch open set containing a.We ↑ ↑ may assume that C = c \F with c ∈ K(P)and F ⊆ P finite, cf. 2.10.It suffices to show that η(c) ∈ C. Note that c ≤ η(c) ≤ a (as a is absorbing) and η(c) ∈ / F (since otherwise x ≤ η(c) ≤ a for some x ∈ F , hence a/ ∈ C,a contradiction). Theorem 4.5 (cf. [12, 1.1(9)]). Let P be an algebraic lattice and Q ⊆ P a closure system. The following conditions are equivalent. (a) Q is an algebraic closure system. (b) Q is the set of Q-absorbing elements. (c) Q is patchclosedin P . (d) The closure map is coherent, hence K(Q)= ϑ(K(P )). (e) Q is sober for σ(P )| . If the equivalent conditions hold then Q is an algebraic lattice. Proof. For the last assertion recall that every algebraic closure system in an algebraic lattice is an algebraic lattice, cf. [8, Proposition I−4.13]. ↓K(P ) (a)⇒(b). Let a = a ∈ P be absorbing for Q. Then ↓K(P ) ↓K(P ) a ≤ η(a)= ϑ a = ϑ(a ) ≤ a P Q (the last inequality since a is absorbing). Thus, a = η(a) ∈ Q. (b)⇒(a). Let D ⊆ Q be up-directed and set a = D. It suffices to show ↓K(P ) that a is absorbing, i.e., a ∈ Q.If c ∈ a then c ≤ D implies c ≤ d for some d ∈ D, and it follows that η(c) ≤ η(d)= d ≤ a, i.e., a is absorbing. (b)⇔(c) is clear by Proposition 4.4. (a)⇒(d). The inclusion map ι : Q → P is a dcpo map and preserves all infima. Thus ϑ = ι is coherent, Corollary 2.13(c), and the equality follows from Proposition 4.2. (d)⇒(c). For z ∈ P \ Q we must find a patch open set C ⊆ P with z ∈ C ↓K(P ) and C ∩ Q = ∅. Since z/ ∈ Q it follows that z< η(z)= η(z) and ↓K(P ) ↓K(P ) u ≤ z for some u ∈ η(z) .As z = z , ϑ preserves suprema and ↓K(P ) u ≤ η(z)= ϑ(z) it follows that ϑ(u) ≤ ϑ(z)= ϑ(z ). By hypothesis ↓K(P ) ϑ(u) ∈ K(Q), hence there is some v ∈ z with ϑ(u) ≤ ϑ(v). The set ↑P ↑P ↓K(P ) v \η(v) is patch-open in P , contains z (since v ∈ z , u ≤ z,and ↑Q ↑Q u ≤ η(u) ≤ η(v)) and is disjoint from Q (as v = η(v) ). Topology of closure systems Page 17 of 33 17 (c)⇒(e). According to [6, Theorem 2.1.3] the patch closed subset Q ⊆ P is a spectral subspace, hence is sober for its inverse topology, which is equal to σ(P )| ,[6, Theorem 7.2.8]. (e)⇒(a). Since σ(P ) is an upper topology its specialization poset is P . inv Pick D ⊆ Q up-directed, hence down-directed in P . Then D is the inv σ(P ) generic point of D ,[6, Proposition 4.2.1(ii)]. Soberness of (Q, σ(P )| ) im- plies D ∈ Q (cf. 2.8(g)), i.e., D = D. P P Q Example 4.6. Let P be an algebraic lattice, Q ⊆ P an algebraic closure system. Theorem 4.5 shows that the inclusion ι : Q → P is a spectral map, hence η is spectral if and only if ϑ is spectral. (Clearly, ϑ spectral implies η = ι ◦ ϑ spectral. If η is spectral and Q ⊆ P is a spectral subspace then the corestriction ϑ : P → Q is spectral.) Both maps are dcpo maps, cf. Remark 3.6(a), hence are continuous for the Scott topology, Theorem 2.11(e). If one of them is also continuous for the coarse lower topology then both are spectral, [6, Theorem 1.4.6]. We show that this need not be the case: Let P be the inverse of the set N ∪{ω}, and define Q = {ω, 0} = {ω} . Note that P is an algebraic lattice and Q is an algebraic closure system. The closure operator η is given by ω → ω and a → 0 otherwise. It is a homomor- phism of bounded lattices and preserves all suprema, but is not an fcpo-map since N = ω and η( N)= ω< 0= η(N). It follows from Corollary 2.14(a) that the closure operator is not a spectral map. In fact, {0}⊆ P is closed for −1 the coarse lower topology, but η ({0})= N is not closed. Example 4.7. Let P be an algebraic lattice and Q ⊆ P a closure system. Then K(Q) ⊆ ϑ(K(P )), Proposition 4.2. We exhibit a closure system that shows how equality can fail. Let P be the totally ordered set 2 · ω + 1, which is an algebraic lattice since every non-limit ordinal is compact. The subset Q = N∪{2·ω} is a closure system in P , but is not algebraic since N = ω< N =2 · ω. Note that Q P Q is isomorphic to ω + 1, hence is an algebraic lattice. The only element that is not compact is 2 · ω. The closure map is given by a → a if a ∈ N and a → 2 · ω otherwise. The element ω +1 ∈ P is not a limit ordinal, hence is compact. But ϑ(ω +1) = 2 · ω is not compact. 5. Closure systems and generating subsets An algebraic lattice P is complete, hence every subset X ⊆ P generates a closure system. We explore connections between properties of the generating set and the closure system. The main results are contained in Theorem 5.2. con They imply, in particular, that the operators P(P ) → P(P ),X → X and P(P ) → P(P ),X →X commute with each other, Remark 5.4. Notation 5.1. The following notation will be used frequently in the rest of the paper. Consider a poset P and a subset X. By default P is equipped with the coarse lower topology. If P is a complete lattice then we set L = X ,and if con P is an algebraic lattice then we also define Y = X and M = Y . P 17 Page 18 of 33 N. Schwartz Algebra Univers. Theorem 5.2. Notation as in 5.1.Let P be an algebraic lattice. (a) M is patch closed in P . con (b) M = L , i.e., M is the set of L-absorbing elements (see Proposition 4.4 and compare with [13, Theorem 4.12]). (c) M is the smallest algebraic closure system in P containing X. (d) K(M ) ⊆ L,hence η (a)= η (a) for a ∈ K(P ). M,P L,P Proof. (a). To show that P \ M is patch open in P,pick z ∈ P \M . Then ↓K(P ) ↓K(P ) z< η (z)= η (z) , hence there is some a ∈ η (z) with M,P M,P M,P ↑Y ↑Y ↑Y ↑Y ↑P a ≤ z. Note that z = η (z) ⊆ a = η (a) ,cf. 3.1. The set a is M,P M,P ↑Y closed and constructible in P and Y ⊆ P is patch closed, hence a is closed ↓K(P ) and constructible in Y . Since z = z and the supremum is up-directed ↑Y ↑Y ↑Y it follows that z = b .The sets b are closed and constructible ↓K(P ) b∈z in Y , the intersection is down-directed for inclusion and is disjoint from the ↑Y ↑Y ↑Y quasi-compact open set Y \ a (since z ⊆ a ). The patch topology of Y ↓K(P ) ↑Y ↑Y is compact, hence there is some b ∈ z with b ⊆ a . One concludes ↑Y ↑Y that η (a)= a ≤ b = η (b). Note that η (b) ≤ z.For, M,P M,P M,P P P otherwise a ≤ η (a) ≤ η (b) ≤ z, contradicting the choice of a.Thus M,P M,P ↑P ↑P ↑M b \η (b) is patch open in P and contains z. Finally, if x ∈ b then M,P ↑P ↑P η (b) ≤ η (x)= x,and b \η (b) is a patch open neighborhood of M,P M,P M,P z disjoint from M . con (b). First we claim that L ⊆ P is closed under finite meets. As ∈ L con con it suffices to prove a∧b ∈ L if a, b ∈ L . The map ∧ : P ×P → P is spectral, Corollary 2.16, hence sends patch closed sets to patch closed sets. As L ⊆ P is closed under infima it follows that ∧ restricts to a map L×L → L. The equality con con con con con con L × L = L × L ,cf. [6, Theorem 2.2.1], implies ∧(L × L )= L . con Next we claim that L is a closure system in P . For, consider any nonempty con subset S ⊆ L and let T ⊆ P be the set of infima F with F ⊆ S finite. Then con T ⊆ L and S = T . The down-directed set T is irreducible, hence T P P is nonempty closed and irreducible in P and has the generic point T,[6, con con Proposition 4.2.1]. Moreover, T ⊆ L is down-directed and contains the generic point of T,[6, Proposition 4.2.6]. To finish the proof, note that X ⊆ L con con con implies Y ⊆ L , hence M ⊆ L ,and L = M follows from item (a). (c) follows from item (b) and Theorem 4.5. ↑M ↑M (d). Pick c ∈ K(M ) and note that the set C = c \η (c) is patch L,M open in M . Assume c<η (c). Then c ∈ C, i.e., C = ∅, and (b) implies L,M ↑L ↑L c \ η (c) = C ∩ L = ∅, a contradiction, cf. 3.1. The final claim follows L,M from Theorem 4.5. Corollary 5.3. Let P be an algebraic lattice and Q ⊆ P a closure system. con (a) Q is an algebraic closure system and an algebraic lattice. con (b) K(Q) ⊆ K(Q ). con con (c) If K(Q)= K(Q ) then Q = Q . con con (d) Q is dense in Q for β(σ(P )).Thus, Q with the Scott topology is the sobrification of (Q, σ(P )| ). con (e) The elements of Q are the suprema in P of up-directed sets in Q. Topology of closure systems Page 19 of 33 17 Proof. (a) follows from Theorem 5.2 and Theorem 4.5. con con ↓K(Q ) con (b). Pick a ∈ K(Q) and write a = a . Then K(Q ) ⊆ Q, con con ↓K(Q ) ↓K(Q ) cf. Theorem 5.2(d), implies a = a . Since a is up-directed con con ↓K(Q ) there is some b ∈ a with a ≤ b, hence a = b ∈ K(Q ). con (c). Pick b ∈ Q and write ↓K(Q) ↓K(Q) ↓K(Q) b = b ≤ a := b = a . con con Q Q ↓K(Q) ↓K(Q) ↓K(Q) ↓K(Q) If a = b then b = a ∈ Q. The inclusion b ⊆ a holds ↓K(Q) ↓K(Q) trivially. If c ∈ a then there is some d ∈ b with c ≤ d ≤ b, i.e., ↓K(Q) c ∈ b . con (d). We may assume that Q = P.The sets x with x ∈ K(P)are abasis of open sets for the Scott topology of P,[8, Corollary II−1.15] or [6, Theorem ↓ ↑ 7.2.8]. Hence the sets a ∩ x , with a ∈ P , x ∈ K(P ) ⊆ Q (Theorem 5.2(d)), ↓ ↑ are a basis for β(σ(P )), see 2.2. The set a ∩ x is nonempty if and only if it ↓ ↑ contains x, if and only if (a ∩ x ) ∩ Q = ∅. – The second assertion follows from 2.3. con (e). If D ⊆ Q is up-directed then D ∈ Q by (a). Conversely, every con con con con ↓K(Q ) ↓K(Q ) ↓K(Q ) con element a ∈ Q is equal to a = a , where a Q P is up-directed and is contained in Q, Theorem 5.2(d). Remark 5.4. Let P be an algebraic lattice. As noted in 3.1 the set C (P)of closure systems and the set A (P ) of algebraic closure systems in P are closure systems in P(P ). The subset A(P ) ⊆ P(P ) of patch closed subsets of P is con a closure system as well. Theorem 4.5 shows that A (P)= A(P ) ∩ C (P ), con and it follows that η ◦ η = η = η ◦ η , A (P ),P(P ) C (P ),P(P ) A (P ),P(P ) A (P ),P(P ) A(P ),P(P ) con con con i.e., X = X for X ∈ P(P ). P P 6. Prime generated closure systems Continuing the study of closure systems and generating sets we consider closure systems generated by sets of prime elements. The notion of prime elements in a poset can be found in [8,I−3.11]. If P be a ∧-semilattice, which will always be the case in our considerations, then p ∈ P is prime if a ∧ b ≤ p implies a ≤ p or b ≤ p for all a, b ∈ P,cf. [8, Proposition I−3.12]. The set of prime elements is denoted by P(P ). A top element, if it exists, is always prime and is called the trivial prime element and P(P ) (which may be empty) is the set of nontrivial prime elements. We show that P(P)and P(P ) are both closed for the b-topology β(τ (P )), Proposition 6.3. A closure system Q ⊆ P is called prime generated if Q = X , where X ⊆ P(Q). Let P be a complete lattice and Q ⊆ P a closure system. It is always true that P(P ) ∩ Q ⊆ P(Q), and the inclusion may be proper. If Q is prime 17 Page 20 of 33 N. Schwartz Algebra Univers. generated then P(P ) ∩ Q = P(Q) if and only if the closure operator is a ∧- homomorphism, Corollary 6.9. Moreover, if Q is prime generated by X then it is isomorphic to the frame O(X ), Theorem 6.13.If P is even an algebraic lattice then it follows from Theorem 6.20 that a prime generated algebraic closure system is coherent if and only if its set of nontrivial prime elements is patch closed in P . Lemma 6.1. Let P be an algebraic lattice. An element p ∈ P is prime if and only if a ∧ b ≤ p implies a ≤ p or b ≤ p for all a, b ∈ K(P ). Proof. If p is prime then the claim holds trivially. Now assume that p is not prime, i.e., there are x, y ∈ P with x ∧ y ≤ p, but x ≤ p and y ≤ p. Since ↓K(P ) ↓K(P ) ↓K(P ) ↓K(P ) x = x and y = y there are a ∈ x and b ∈ y with a, b ≤ p, but a ∧ b ≤ x ∧ y ≤ p. Example 6.2. (a). For any set S the power set P(S) is an algebraic frame, and the set of nontrivial prime elements is equal to {T ⊆ S ||S\T | =1}. (b). Let K be a field with at least 3 elements and let V be a vector space with dim V ≥ 2. Let U (V ) be the set of subspaces, which is an algebraic closure system in P(V ), hence is an algebraic lattice. The compact elements are the finite dimensional subspaces. Thus U (V ) is always an arithmetic algebraic lattice. Coherence holds if and only if dim V is finite. There are no nontrivial prime elements in U (V ). (c). Let P be a ∧-semilattice and X ⊆ P a dense subset. Then X is totally ordered if and only if P is totally ordered, if and only if P = P(P ). (d). Consider ∧-semilattices P and Q with top elements. One checks that P(P × Q)= P(P ) ×{ }∪{ }× P(Q). Thus the projection maps π Q P P and π , as well as their right adjoint maps, send prime elements to prime elements, 2.15.Moreover, ∧ : P × P → P, (a, b) → a ∧ b maps prime elements to prime elements. Proposition 6.3. Let P be a ∧-semilattice with the coarse lower topology. Then P(P ) and P(P ) are closed in P for β = β(τ (P )).Thus, P(P ) and P(P ) are sober if P is complete. Proof. Pick x ∈ P \ P(P ). There are a, b ∈ P with a, b ≤ x and a ∧ b ≤ x.The ↑ ↑ ↑ set (a ∧ b) \(a ∪ b )isa β-neighborhood of x and is disjoint from P(P ). – The subset P ⊆ P is τ (P )-open, hence is β-closed. Thus P(P ) = P(P ) ∩ P is β-closed. – If P is complete then τ (P ) is sober, Proposition 3.8(b). Hence P(P)and P(P ) are sober by 2.3. Example 6.4. Let P be a complete lattice. Then the infimum of a down- directed set of prime elements is prime by Proposition 6.3. We show that the supremum of an up-directed set of prime elements need not be prime. Let P = N ∪{ω}∪{a, b, c} with the following partial order: The set N ∪{ω} of ordinals carries the natural total order and all its elements are smaller than a, b, c. Moreover we define a, b < c and assume that a, b are incomparable. Then P is an algebraic lattice, N ⊆ P(P)and N = ω/ ∈ P(P ). Topology of closure systems Page 21 of 33 17 Lemma 6.5. Let P and Q be ∧-semilattices, ϕ : P → Q aposet mapwithleft ∗ ∗ adjoint ϕ .If ϕ is a ∧-homomorphism then ϕ maps prime elements to prime elements. ∗ ∗ ∗ Proof. If p ∈ P(P)and b ∧ b ≤ ϕ(p) then ϕ (b) ∧ ϕ (b )= ϕ (b ∧ b ) ≤ p, ∗ ∗ hence ϕ (b) ≤ p or ϕ (b ) ≤ p, hence b ≤ ϕ(p)or b ≤ ϕ(p). Remark 6.6. Let P be a ∧-semilattice with top element and Q ⊆ P a ∧- subsemilattice, e.g., a closure system. The inclusion P(P )∩Q ⊆ P(Q) is obvious and may be proper. For example, consider a power set P(S) where |S|≥ 3, cf. Example 6.2(a). There are distinct elements A, B ∈ P(S) \ P(P(S)). We define Q = {A, B} = {A ∩ B, A, B, S}. Then A, B ∈ P(Q)\P(P(S)). P(S) Now assume that Q is a closure system. If the closure map ϑ = ι (equivalently, the closure operator, cf. Remark 3.6)isa ∧-homomorphism then P(P ) ∩ Q = P(Q) by Lemma 6.5. But note that the equality P(P ) ∩ Q = P(Q) may be true without η being a ∧-homomorphism. For an example let K be a field and V a vector space, U (V ) ⊆ P(V ) the algebraic closure system of subspaces, Example 6.2(b). Then P(P(V )) ∩ U (V )= {V } = P(U (V )), but the closure operator is not a ∧-homomorphism. For, consider two disjoint gen- erating sets A and B of a nontrivial subspace U ⊆ V . Then η(A)= U = η(B), but η(A ∩ B)= {0}. Proposition 6.7. Let P be a complete lattice, X asubsetand Q = X .Then ↑X ↑X ↑X ↑X the inclusions a ∪ b ⊆ (η (a) ∧ η (b)) ⊆ (a ∧ b) hold for all Q,P Q,P a, b ∈ P,and (a) both inclusions are equalities if and only if X ⊆ P(P ); (b) the first inclusion is an equality if and only if X ⊆ P(Q); (c) the second inclusion is an equality if and only if η is a ∧-homomor- Q,P phism. Proof. The inequalities η (a) ∧ η (b) ≤ η (a),η (b) imply Q,P Q,P Q,P Q,P ↑X ↑X ↑X ↑X ↑X a ∪ b = η (a) ∪ η (b) ⊆ (η (a) ∧ η (b)) . Q,P Q,P Q,P Q,P The second inclusion follows from a ∧ b ≤ η (a) ∧ η (b). Q,P Q,P (a) is clear by the definition of prime elements. ↑X ↑X ↑X (b). The equality a ∪ b =(η (a) ∧ η (b)) holds for all a, b ∈ P Q,P Q,P ↑X ↑X ↑X if and only if the equality c ∪ d =(c ∧ d) holds for all c, d ∈ Q,ifand only if X ⊆ P(Q), cf. item (a). (c). The second inclusion is an equality if and only if ↑X ↑X ↑X η (a ∧ b) =(a ∧ b) =(η (a) ∧ η (b)) , Q,P Q,P Q,P and the claim follows from 3.1. Example 6.8. Notation as in Proposition 6.7. We show that each inclusion in Proposition 6.7 can be proper while the other one is an equality. In the first example of Remark 6.6 the second inclusion is proper and the first one is an equality. For the other example let P be a complete lattice such that P(P)is not totally ordered and let Q = P(P ) . Both inclusions are equalities, and η is a ∧-homomorphism. Now consider the trivial generating set Q of Q. Q,P 17 Page 22 of 33 N. Schwartz Algebra Univers. Then P(Q) is a proper subset of Q, cf. Example 6.2(c), hence the first inclusion is proper, whereas the second one is an equality since η does not depend Q,P on the particular generating set. Corollary 6.9 (cf. [12, 1.1(9)]). Let P be a complete lattice, Q a closure system, Q = P(Q) . Then the following conditions are equivalent: (a) η is a ∧-homomorphism. (b) P(P ) ∩ Q = P(Q). (c) P(Q) ⊆ P(P ). Proof. (a)⇒(b). See Remark 6.6.(b)⇒(c) is trivial. (c)⇒(a) follows from Proposition 6.7. Corollary 6.10. Let P be an algebraic lattice and use the notation of 5.1.Then (a) P(M ) ∩ L = P(L). (b) If X ⊆ P(L) then η is a ∧-homomorphism and P(M ) ⊆ Y . L,M Proof. (a). The inclusion P(M ) ∩ L ⊆ P(L) holds by Remark 6.6. For the other inclusion pick c ∈ L\P(M)and a, b ∈ K(M ) with a, b ≤ c, a ∧ b ≤ c, Lemma 6.1. Since K(M ) ⊆ L, Theorem 5.2(d), we see that c/ ∈ P(L). (b). Proposition 6.7 and item (a) show that η is a ∧-homomorphism. L,M ↑M ↑M For the second assertion pick b ∈ M \Y . There is a constructible set c \F in M (with c ∈ K(M)and F ⊆ K(M ) finite) containing b and disjoint from X, ↑X ↑M ↑X i.e., c ⊆ F .As K(M ) ⊆ L (Theorem 5.2(d)) it follows that c = c . ↑M ↑M Thus F ≤ c ≤ b, hence b ∈ ( F ) \F , i.e., b/ ∈ P(M ). M M Proposition 6.11. Assume P be a complete lattice and consider X, Y ⊆ P(P ) . β β Then X = Y if and only if X = Y where β = β(τ (P )). P P β β β β Proof. If X = Y then X = X = Y = Y , Proposition 3.7(a). P P P P Conversely, suppose X = Y . We may assume that X and Y are β- P P closed, cf. loc.cit., and claim that X = Y .If y ∈ Y \X then y ∈X implies ↑X ↑X y = y , where y = ∅. Since X is β-closed there is a finite nonempty set ↑P ↑P F ⊆ P such that y \ F is a β-open neighborhood of y and is disjoint from ↑X ↑X ↑X X. But then y ⊆ F =( F ) (whereweuse X ⊆ P(P )). It follows ↑P that F ≤ y,and y ∈ P(P ) yields y ∈ F , a contradiction. Corollary 6.12. Let P be a complete lattice, X ⊆ P(P ) and Q = X .Then P(Q) is the closure of X for β = β(τ (P )), i.e., P(Q) is the sobrification of X (cf. 2.3). Proof. Proposition 6.7 and Corollary 6.9 imply P(Q) = P(P ) ∩ Q. Both P(P ) ⊆ P and Q ⊆ P are closed for β, cf. Proposition 6.3 and Proposition 3.7(a), hence P(Q) ⊆ P is closed as well. Thus P(Q) = X follows from Proposition 6.11 and Q = X = P(Q) . P P Theorem 6.13. Let P be a complete lattice generated by X ⊆ P(P ) . ↑X (a) If A ∈A(X) then A = a where a = A. P Topology of closure systems Page 23 of 33 17 (b) P is isomorphic to the spatial frame O(X ). ↑X Proof. (a). Trivially, A ⊆ a .Now pick p ∈ X\A. Since A is closed for ↑P ↑P ↑P τ (P )| there is a finite set F ⊆ P with A ⊆ F ⊆ ( F ) and p/ ∈ F . It follows that F ≤ a, hence F ≤ p (since p ∈ P(P )), and we see that P P a ≤ p. ↑X (b). The map γ : P →O(X, ↑(X, P )),a → X\a is an isomorphism X,P of posets, Lemma 3.5. Thus it suffices to show O(X, ↑(X, P ))) = O(X). The inclusion O(X, ↑(X, P )) ⊆O(X) holds trivially, Example 3.3, and equality follows from item (a). Remark 6.14. Theorem 6.13 is a more precise version of [8,I−3.15] where it is shown that a complete lattice is a frame if it is prime generated and that continuous (in particular: algebraic) lattices are frames if and only they are prime generated. Prime generated frames are exactly the spatial frames, [11,p. 43]. If X ⊆ P(P ) is generating then P(P ) is the sobrification of X, Corollary 6.12, and the canonical map O(P(P ) ) →O(X) is an isomorphism. Remark 6.15. Let P be a complete lattice and define N (P ) to be the set of prime generated closure systems Q ⊆ P such that η is a ∧-homomorphism. Q,P If Q ∈ N (P ) then Q = P(Q) and P(Q) ⊆ P(P ) , Corollary 6.9.Thus Proposition 6.11 shows that N (P ) →A(P(P ) ,β),Q → P(Q) and A(P(P ) ,β) → N (P ),X →X are mutually inverse isomorphisms of posets. Each ele- ment of N (P ) is a spatial frame by Theorem 6.13. Example 6.16. Consider a spatial (= prime generated) frame P.For each Q ∈ N (P ) the closure operator η : P → P is a nucleus and Q is a spatial frame. Q,P Conversely, if ν : P → P is a nucleus such that ν(P ) is a spatial frame then ν(P ) ∈ N (P ), Corollary 6.9. Identifying Q ∈ N (P ) with the nucleus η Q,P we consider N (P ) as a subset of the assembly (= the set of all nuclei), [11, p. 51 ff], [18], namely the set of nuclei with spatial image. In general this is a proper subset. We show how one can construct spatial frames P and nuclei ν : P → P such that ν(P ) is not spatial, hence is not in N (P ). Let X be a localic space,cf. [17, p. 1163] and [16, Section 3] (where localic spaces were introduced under the name locales), such that the frame K(X) does not have enough localic points, cf. [11, p. 43], [16, p. 27], i.e., is not spatial and not prime generated. For example, X could be an infinite extremally disconnected Boolean space without isolated points, [6, Examples 9.5.4(ii)]. (Then K(X) does not have any localic points at all.) The frame con P = O(X) is algebraic and coherent. The map ν : P → P, O → O is a nucleus with ν (P)= K(X) and is called the natural nucleus for X,[16,p. ◦ ◦ 13], [17, p. 1163]. It follows that P(P ) ∩ K(X)= P(K(X)), Remark 6.6.As K(X) is not prime generated it does not belong to N (P ). Proposition 6.17. Let P and Q be complete lattices generated by X ⊆ P(P ) and Y ⊆ P(Q) .Let ϕ : P → Q be a -homomorphism with ϕ(X) ⊆ Y 17 Page 24 of 33 N. Schwartz Algebra Univers. and let f : X → Y be the restriction of ϕ. Then the following diagrams are commutative: ϕ ϕ Q P P Q γ γ γ γ Y,Q X,P X,P Y,Q O(f ) O(f ) O(Y ) O(X) O(X) O(Y ) Proof. Note that the left adjoint ϕ exists by 2.6. It suffices to prove commu- tativity for the diagram on the left. Uniqueness of right adjoints implies the claim for the other diagram. Theorem 2.11(c) shows that ϕ is continuous for the coarse lower topology, hence f is continuous as well. For each b ∈ Q the ↑Y set γ (b)= Y \b ⊆ Y is open, hence Y,Q ↑Y −1 ↑Y O := O(f )(Y \ b )= X \ f (b ) ⊆ X ↑X is open and equals X \ z = γ (z), where X,P z = X \ O = {x ∈ X | b ≤ f (x)}, P P Theorem 6.13(a). It remains to show that ϕ (b)= z. Recall from 2.6 that ϕ (b)= {a ∈ P | b ≤ ϕ(a)}, hence the inclusion {x ∈ X | b ≤ ϕ(x)= f (x)}⊆{a ∈ P | b ≤ ϕ(a)} yields ϕ (b) ≤ z. To prove equality we show that b ≤ ϕ(a) implies z ≤ a. ↑X ↑X ↑X As a = a we have ϕ(a)= ϕ(a )= f (a ). Thus b ≤ ϕ(a) P Q Q ↑X ↑X implies b ≤ f (x) for all x ∈ a , i.e., a ⊆{x ∈ X | b ≤ f (x)}, hence ↑X z ≤ a = a. We continue with the notation and hypotheses of Proposition 6.17.It follows from Theorem 6.13 that P and Q are frames and O(f ) (U)isthe −1 largest V ∈O(Y ) with f (V ) ⊆ U . Accordingly ϕ(a) is the largest element ∗ ∗ b ∈ Q with ϕ (b) ≤ a. The composition ϕ ◦ ϕ is a closure operator, 2.6(a). In fact, it is a nucleus of the frame Q since ϕ is a frame homomorphism and ϕ preserves all infima. If P and Q are algebraic lattices then they are spectral spaces and ϕ and ϕ are maps between spectral spaces. We ask whether they are spectral maps. We know from 2.6 and Theorem 2.11(c), (e) that ϕ is coarse lower continuous and ϕ is Scott continuous. Proposition 6.18. (a) Let X ⊆ P and Y ⊆ Q be patch closed. Then ϕ is spectral if and only if f is spectral. (b) ϕ is spectral if and only if for each a ∈ K(P ) there is a smallest element b ∈ Q with a ≤ ϕ (b). If this is the case then b ∈ K(Q). Proof. (a). If ϕ is a spectral map then its restriction f is trivially spectral. Conversely assume that f is spectral. By Corollary 2.13(c) it suffices to show that ϕ,or O(f ) , is a dcpo map. Recall from Example 3.2 that O(f ) ◦O(f)is ∗ ∗ a closure operator and O(f ) ◦O(f ) is a kernel operator. For any up-directed ∗ Topology of closure systems Page 25 of 33 17 set U⊆ O(X) the inclusion O(f ) (U ) ⊆O(f ) ( U ) holds trivially, and ∗ ∗ we must prove equality. Assume the inclusion is proper. Then there is some W ∈ K(Y ) with W ⊆ O(f ) (U ), but W ⊆O(f ) ( U ). It follows that ∗ ∗ −1 f (W)= O(f )(W ) ∈ K(X)and O(f )(W ) ⊆O(f ) ◦O(f ) ( U ) ⊆ U . Since U is up-directed there is some U ∈U with O(f )(W ) ⊆ U . But then W ⊆O(f ) ◦O(f )(W ) ⊆O(f ) (U ), a contradiction. ∗ ∗ ↑P (b). Pick a ∈ K(P ), i.e., a is closed and constructible, cf. 2.10, and let b ∈ Q be the smallest element with a ≤ ϕ (b). It suffices to show that b ∈ K(Q) ∗ −1 ↑P ↑Q ↓K(Q) since then (ϕ ) (a )= b is closed and constructible. As b = b ∗ ∗ ∗ ↓K(Q) and ϕ preserves all suprema it follows that a ≤ ϕ (b)= ϕ (b ). The ↓K(Q) supremum is up-directed and a ∈ K(P ), hence there is some c ∈ b with a ≤ ϕ (c). Minimality of b implies c = b. ∗ ∗ −1 ↑P Now assume ϕ is spectral. For a ∈ K(P)the set(ϕ ) (a ) is closed ↑Q and constructible in Q, hence is equal to G with G ⊆ K(Q) finite, cf. 2.10. ∗ ∗ ∗ It follows that a ≤ ϕ (y) for all y ∈ G,thus a ≤ ϕ (G)= ϕ ( G) (note P Q that ϕ is a frame homomorphism, Proposition 6.17). We define b = G and claim that this is the smallest c ∈ Q such that a ≤ ϕ (c). So, pick any z ∈ Q ∗ ∗ −1 ↑P ↑Q ↑Q with a ≤ ϕ (z). Then z ∈ (ϕ ) (a )= G ⊆ b , i.e., b ≤ z. Example 6.19. We continue with Proposition 6.18(b) and assume X ⊆ P and τ (P ) Y ⊆ Q are patch closed. Since P = X = P(P ) and X = X it P P follows from Proposition 6.11 that X = P(P ) . The same holds for Y and Q. We exhibit a few examples where O(f ) is spectral. The condition in Proposition ◦ ◦ 6.18(b) says that for each U ∈ K(X) there is a smallest V ∈ K(Y ) with −1 U ⊆ f (V ). Since f is continuous and U is quasi-compact it follows that f (U ) ⊆ Y is quasi-compact, hence Gen(f (U )) is patch closed in Y and is the intersection of all open sets containing f (U ), [6, Theorem 4.1.5]. So, there is a smallest open set containing f (U ) if and only if Gen(f (U )) is open, if and only if Gen(f (U )) is quasi-compact open. Here are a few situations where this is the case. Obviously, the condition holds if f is an open map. Next assume that Y is totally ordered. Then Q is totally ordered as well and is equal to Y , see Example 6.2(c). If C ⊆ Y is a nonempty quasi-compact set then C has a largest element c, and Gen(C)=Gen(c), [6, Proposition 1.6.1]. In particular, if U ∈ K(X) is nonempty then f (U ) has a largest element. The set Gen(c)is open if and only if c is the lower point of a jump or is the closed point of Y . Thus, O(f ) is spectral if and only if for each U ∈ K(X) the largest point of f (U ) is the closed point of Y or is the lower point of a jump. Now let Y be any spectral space, define X = Y to be the patch space, con and f =con : X → Y the identity map, [6, 1.3.24]. Then C ⊆ X is quasi- compact open if and only if it is constructible in Y .Thus, O(f ) is spectral if and only if Gen(C) ⊆ Y is open for each constructible set C ⊆ Y . Equivalently, in Y the closure of every constructible set is constructible, i.e., Y is a Heyting inv inv space, [6, Definition 8.3.1]. 17 Page 26 of 33 N. Schwartz Algebra Univers. Theorem 6.20. Notation as in 5.1.Let P be an algebraic lattice, X ⊆ P(P ). (a) If P is arithmetic then P(P ) is patchclosedin P . (b) If P is coherent then P(P ) is patch closed in P . (c) Suppose that X is patchclosedin P.Then L is an arithmetic algebraic frame and is patch closed in P . Moreover, P(L)= P(P ) ∩ L = X . (d) Suppose that X is patch closed in P and is contained in P(P ) .Then L is a coherent algebraic frame and is patch closed in P . Moreover, P(L) = P(P ) ∩ L = X. ↑P ↑P Proof. (a). The sets a \ G , with a ∈ K(P)and G ⊆ K(P ) finite, are a basis of open sets for the patch topology of P , 2.10.If x ∈ P \ P(P ) then we ↑P ↑P ↑P ↑P must find such a and G with x ∈ a \G and a \G ∩ P(P)= ∅.As x is not prime there are y, z ∈ K(P ) with y ∧ z ≤ x and y, z ≤ x, Lemma 6.1. Thus, a = y ∧ z and G = {y, z} meet the requirements. (b). Item (a) shows that P(P ) ⊆ P is patch closed. It follows from K(P ) that P is quasi-compact open, 2.10, hence P(P ) = P(P ) ∩ P is patch closed. (c). Theorem 5.2(a), Theorem 4.5 and Theorem 6.13 show that L = M is patch closed in P , is an algebraic lattice and isomorphic to the frame O(X ). To see that L is arithmetic we pick a, b ∈ K(L) and claim that a ∧ b ∈ K(L). ↑X ↑X ↑L ↑L The sets a ,b ⊆ X are closed and constructible since a ,b ⊆ L are ↑X ↑X ↑X closed and constructible. Proposition 6.7 implies (a ∧ b) = a ∪ b ↑X ↑X (since X ⊆ P(L)). As a ∪ b is closed and constructible it follows that a ∧ b ∈ K(L). The claim about the prime elements follows from Corollary 6.9 and Corollary 6.10. (d). By item (c) it suffices to show that ∈ K(L). We identify L = O(X), Theorem 6.13, and note that = X ∈ K(O(X)). O(X) Corollary 6.21. Let P be an algebraic lattice and Q a prime generated algebraic closure system. Then Q is coherent if and only if P(Q) ⊆ P is patch closed. Proof. By Theorem 4.5 a subset S ⊆ Q is patch closed in Q if and only if it is patch closed in P . Thus, Theorem 6.20(b), (d) implies that Q is coherent if and only if P(Q) ⊆ P is patch closed. Corollary 6.22. Notation as in 5.1.Let P be an algebraic lattice and assume X ⊆ P(P ) . Consider the following conditions about M . (a) M is arithmetic. (b) M is coherent. (c) Y = P(M ). (d) Y = P(M ) . Then (b)⇔(d)⇒(a)⇔(c) and the conditions are all satisfied if there is a patch closed subset Z ⊆ P with X ⊆ Z ⊆ P(P ) . Proof. (a)⇒(c). Theorem 6.20(a) shows that Y ⊆ P(M ), and P(M ) ⊆ Y holds by Corollary 6.10. The other implications, as well as the final assertion, are clear by Theorem 6.20. Topology of closure systems Page 27 of 33 17 We continue with the hypotheses of Corollary 6.22. Assume that M is not arithmetic. We ask how close the elements of Y are to being prime. Proposition 6.23. Notation as in 5.1.Let P be an algebraic lattice, X ⊆ P(L). For z ∈ M the following conditions are equivalent: (a) z ∈ Y . ↑M (b) If F ⊆ K(M ) is nonempty and finite and z/ ∈ F then F z. Proof. (a)⇒(b). Suppose z ∈ P(M)and F ⊆ M is finite and nonempty with ↑M z/ ∈ F . Then F ≤ z, hence F z,[8,I−1.2]. Now assume that M M ↑M z/ ∈ P(M ) and that there is a finite subset F ⊆ K(M ) with z/ ∈ F , but F z. (Note that F = ∅ since otherwise = F ≤ z, hence z = M M ↓K(M ) ∈ P(M ).) As z = z and the supremum is up-directed there is ↓K(M ) ↑M ↑M ↑M ↑M some t ∈ z with F ≤ t, hence z ∈ t \F . The set t \F is con patch open in M,cf. 2.10, and is disjoint from X, hence z/ ∈ Y = X ,a contradiction. (b)⇒(a). As P(M ) ⊆ Y , Corollary 6.10, we assume z/ ∈ P(M ). Then ↑M ↑M ↑M there are u, v ∈ K(M ) with z ∈ (u ∧ v) \(u ∪ v ), Lemma 6.1.Wehave to show that C ∩ X = ∅ for any constructible set C ⊆ M with z ∈ C. It suffices ↑M ↑M to consider basic constructible sets, say C = c \F with c ∈ K(M)and F ⊆ K(M ) finite, 2.10. Then H = F ∪{u, v}⊆ K(M ) is finite and has the ↑M property that H ≤ u ∧ v ≤ z and z/ ∈ H . The hypothesis says that ↓K(M ) H z.Now c ∈ z implies c z,[8,I−1.2], and we conclude that ↑X H ≤ c. Since K(M ) ⊆ L, Theorem 5.2(d), we have c = c ∈ L and M M ↑X H ∈ L. Hence there is some p ∈ c with H ≤ p, and this implies M M ↑X ↑X p ∈ c \H ⊆ C ∩ X. The following examples illustrate the preceding results. Example 6.24. For any set S the power set P(S) is an algebraic frame. The compact elements are the finite subsets, hence P(S) is arithmetic, but is coher- ent only if S is finite. The nontrivial prime elements are described in Example 6.2(a). We identify S with P(P(S)) via s → S \{s}, hence S = P(P(S)). The coarse lower topology of P(S) restricts to the discrete topology on S,and S ∪{S} is a spectral subspace of P(S), Theorem 6.20(a). Specialization in S is given by s ≤ S for all s ∈ S. The arithmetic algebraic frame P(S)is isomorphic to O(S), S with the discrete topology. Example 6.25 (cf. [11, p. 50, 2.4(a)]). Let P be an algebraic lattice, a ∈ P . Obviously the subset a ⊆ P is an algebraic closure system. The closure op- erator η : P → P is given by x → a ∨ x. This is always a dcpo homo- morphism, but is a -homomorphism only if ⊥ = a.(Forif ⊥ <a then P P η( ∅)= a> ⊥ = η(∅).) The algebraic lattice P is a spectral space for P P the coarse lower topology and a is a closed subset, hence is a spectral space as well, and the inclusion ι : a → P is a spectral map. ↑ ↑ Set Y = P(P ) and X = P(a ) . The inclusion Y ∩ a ⊆ X is obvious. a a Let η be a ∧-map. Then Y ∩ a = X , Remark 6.6,and X ⊆ Y is a closed a a subset. Now assume that P is distributive, equivalently a prime generated 17 Page 28 of 33 N. Schwartz Algebra Univers. algebraic frame, [8, Theorem I−23.15]. Then P O(Y )and η is a nucleus. ↑X ↑Y ↑ ↑Y ↑X a a The equality b = b holds for each b ∈ a , hence b = b = b , P a and a is prime generated as well. Next consider the compact elements. It follows from Proposition 4.2 and ↑ ↑ ↑ Theorem 4.5 that K(a )= ϑ(K(P )). It is trivially true that K(P )∩a ⊆ K(a ), and equality holds if and only if a ∈ K(P ). For, assume a ∈ K(P ) and pick c ∈ K(a ), D ⊆ P up-directed with a ≤ c ≤ D. There is some d ∈ D with ↑ ↑ ↑ a ≤ d.Thus, d ∩ D ⊆ a is up-directed and c ≤ D = d ∩ D.As P a ↑ ↑ c ∈ K(a ) there is some x ∈ d ∩ D with c ≤ x, hence c ∈ K(P ). Conversely assume a/ ∈ K(P ). Then ⊥ ↑ = a is compact in a , but not in P . Consider the diagrams of Proposition 6.17 and assume P is prime gen- erated. In general X ⊆ a and Y ⊆ P will not be spectral subspaces. Let f : X → Y be the inclusion. a a ϑ ι a a ↑ ↑ P a a P γ γ γ γ Y,P ↑ ↑ Y,P X ,a X ,a a a O(f ) O(f ) a a ∗ O(Y ) O(X ) O(X ) O(Y ) a a As noted above, ι is always spectral. But ϑ need not be spectral. For an a a example, let Y be an infinite set with the cofinite topology and set P = O(Y ). Then P is a prime generated frame and is algebraic since Y is a Noetherian space, i.e., K(P)= K(Y )= O(Y )= P,[6, Proposition 8.1.5]. Pick x, y ∈ Y , set a = Y \{x, y},b = Y \{y}∈ P . Then b ∈ K(P ) ∩ a and b = a ∨ c = ϑ (c) for any c ∈ P with x ∈ c and y/ ∈ c. Clearly, there is no smallest such c ∈ P . On the other hand, one checks that ϑ is spectral if Y is Boolean and a ∈ K(P ). Example 6.26 (cf. [11, p. 50, 2.4(b)]). Let P be an algebraic lattice, pick a ∈ P ↓ ↓ and consider the principal downset a . The inclusion map ε : a → P is a - morphism, a ∧-morphism and an fcpo map, however not a -morphism since a = ε ( ∅) < = ∅ if a< . The map μ : P → a ,x → a ∧ x a ↓ P P a P ↓K(P ) ↓ ↓ is right adjoint to ε , 2.6. One checks that a = K(a ), hence a is an algebraic lattice as well. ↓ ↓ The subset a ⊆ P is patch closed and τ (a )= τ (P )| ↓.Thus, ε is a spectral map. It is claimed that μ is spectral as well, i.e., is coarse lower −1 continuous and Scott continuous. We show that μ (C) ⊆ P is closed and constructible if C ⊆ a is closed and constructible. By 2.10 it suffices to con- ↑a ↓ −1 ↑P sider sets C = c with c ∈ K(a ). But then c ∈ K(P)and μ (C)= c is closed constructible in P . Theorem 2.11(e) yields the somewhat surprising fact that μ is a dcpo map without any assumption about distributivity, also see [6, Corollary 4.2.9]. This can also be proved using the following order-theoretic arguments. If D ⊆ P is up-directed with z = D then μ(D) ⊆ a is up-directed as well. With t = μ(D)wehavetoshow μ(z)= t. It is clear that t ≤ μ(z)and we assume t<μ(z). Then there is some c ∈ K(a ) with c ≤ t and c ≤ μ(z). Topology of closure systems Page 29 of 33 17 Hence K(a ) ⊆ K(P)and c ≤ μ(z) ≤ z imply c ≤ d for some d ∈ D. But then c = μ(c) ≤ μ(d) ≤ t, a contradiction. Now let P be an algebraic frame. Then a is a frame as well and μ is a surjective frame homomorphism. The right adjoint μ : a → P exists and is given by b → (a → b), cf. 2.6 and [11, p. 8]. The map λ = μ ◦ μ : P → P P ∗ is a nucleus, [11, p. 50, 2.4(b)], and Q := λ(P ) is a frame and a closure system in P (in general not algebraic). The corestriction λ : P → Q of λ and the inclusion λ : Q → P form the adjoint pair (λ ,λ ). The maps ∗ a ∗ ↓ ∗ ↓ μ ◦ λ : Q → a (restriction of μ)and λ ◦ μ : a → Q (corestriction of μ ) ∗ a ∗ a ∗ are mutually inverse frame isomorphisms. Thus Q is an algebraic frame as well and the spectral spaces a and Q are homeomorphic. The topologies τ (Q ) a a and τ (P )| coincide by Remark 3.6.But Q ⊆ P is a spectral subspace if Q a and only if the closure system is algebraic, Theorem 4.5. By [8,TheoremI−3.15] the algebraic frames P , a and Q are prime generated. As λ is a nucleus it follows that Q ∈ N (P ), Remark 6.15,and P(P ) ∩ Q = P(Q ), Remark 6.6. Set Y = P(P ) and U = P(Q ) = Y ∩ Q a a a a a and let g : U → Y be the inclusion. Then P O(Y )and a Q O(U ), a a a a Theorem 6.13. In general P(P ),Y, P(Q ),U are not spectral subspaces of P . a a However, see Theorem 6.20. Compare the following considerations with [13, Lemma 4.6]. We claim that μ maps prime elements to prime elements. To prove this pick p ∈ P(P ). ↓ ↑ ↓ If a ≤ p then μ(p)= a ∈ P(a ). Now assume p ∈ P \a and b, c ∈ a with b ∧ c ≤ μ(p). Then b ∧ c ≤ p, which implies b ≤ p or c ≤ p, hence b ≤ μ(p) or c ≤ μ(p). Thus, μ(p) ∈ P(a ) , proving the claim. The isomorphisms μ ◦ λ ∗ ↓ and λ ◦ μ restrict to the mutually inverse maps U → P(a ) ,p → a ∧ p and ∗ a ↓ ↑ P(a ) → U ,q → (a → q). It follows that U = Y \a is open in Y .Thus, a P a the open subspaces {U | a ∈ P } cover Y .If a ∈ K(P ) then a ∈ K(a ), hence P(a ) and U are quasi-compact. As P is algebraic the space Y is locally quasi-compact in the sense that the quasi-compact open sets are a basis of open sets, cf. [8, Definition 0−5.9]. Finally consider the diagrams of Proposition 6.17. λ ∗ P Q Q P a a γ γ γ γ Y,P U ,Q U ,Q Y,P a a a a O(g ) O(g ) a a ∗ O(Y ) O(U ) O(U ) O(Y ) a a As λ ◦ (λ ◦ μ )= μ it follows that λ is a spectral map if and only if μ is ∗ ∗ ∗ ∗ ∗ a spectral map, if and only if μ is coherent, see Corollary 2.14. In general this is not the case. However, assume that P is arithmetic and a ∈ K(P ). Then ↓ ↓ ↓ c ∈ K(P ) implies μ(c) ∈ K(P ) ∩ a = K(a )and a is a coherent algebraic frame, hence U is a spectral space by Theorem 6.20. In fact, {U | a ∈ K(P )} a a is closed under finite unions and intersections and every quasi-compact open subset is spectral, hence Y is a locally spectral space (i.e., the open spectral subspaces are a basis). 17 Page 30 of 33 N. Schwartz Algebra Univers. ∗ ∗ ∗ On the other hand, the equality λ =(λ ◦ μ ) ◦ μ shows that λ is spectral if and only if μ is spectral, which is always true as shown above, also see Proposition 6.18(b). 7. Closure systems in coherent algebraic frames Finally we consider closure systems in frames. We start with a couple of results that are special cases of (or follow easily from) the previous sections. The main result of the section is Theorem 7.1 which says that, for a coherent algebraic frame P , the set A (P ) ∩ N (P ) of closure systems, cf. 3.1 and Remark 6.15, is a closure system in P(P ). Let P be a frame and Q ⊆ P a closure system. In general the closure operator is not a ∧-homomorphism. For an example let P be any frame that is not totally ordered. Pick a ∈ P \P(P ), cf. Example 6.2(c), and define Q = {a} = {a, }. Then Q is an algebraic closure system in P , is prime- P P generated since Q = P(Q), and is trivially coherent. But η is not a ∧- Q,P homomorphism, which follows from Remark 6.6 or can be checked directly. Now assume that the closure operator is a ∧-homomorphism. Then η is a nucleus, hence Q is a frame, ϑ : P → Q is a frame homomorphism, and P(P ) ∩ Q = P(Q), Remark 6.6. However Q need not be prime generated (Example 6.16)and P(Q) and P(P ) may both be empty. As in Remark 6.15 let N (P ) be the set of prime generated closure systems Q such that η is a nucleus. The poset isomorphism Q,P N (P ) →A(P(P ) ,β),Q → P(Q) shows that P(P ) O(P(P ) ) is the largest element of N (P ). If P is a continuous frame then P = P(P ) ,[8, Theorem, I−3.15], and P is algebraic if and only if K(P(P ) )isabasisofopensetsfor P(P ) . If P is a coherent algebraic frame then P(P ) generates P and is a patch closed subset, cf. Theorem 6.20(b). Consider A (P )∩N (P ), the set of algebraic closure systems in N (P ). We claim that each Q ∈ A (P ) ∩ N (P ) is a coherent algebraic frame. First note that Q ⊆ P is patch closed by Theorem 4.5. Being prime generated, Q is a frame, Theorem 6.13, and the closure operator η is Q,P a nucleus since it is a ∧-homomorphism. Corollary 6.9 implies that P(P ) ∩Q = P(Q) , which is patch closed in P,and Q is coherent by Theorem 6.20(d). But N (P ) may contain coherent algebraic frames that are not algebraic closure systems in P . Examples can be constructed as follows: Let Y be a spectral space and X a spectral space that is a subspace of Y , but not a spectral subspace, see Example 4.7 or [6, Example 2.1.2], and let e : X → Y be the inclusion map. Then O(e): O(Y ) →O(X) is a surjective homomorphism of prime generated coherent algebraic frames and we can identify X = P(O()) , resp. Y = P(O(Y )) , via the homeomorphisms X → P(O(X)) ,x → X\x , resp. Y → P(O(Y )) ,y → Y \y . The right adjoint O(e) exists and O(e) ◦ ∗ ∗ O(e): O(Y ) →O(Y ) is a closure operator by Example 3.2, even a nucleus since O(e) is a frame homomorphism and O(e) is a -morphism, 2.6.Let Q be ∗ Topology of closure systems Page 31 of 33 17 the image of the nucleus. Since O(e) : O(X) → Q is an isomorphism it follows that Q is prime generated and belongs to N (O(Y )). But Proposition 6.18(a) shows that O(e) is not a spectral map (as e is not a spectral map), hence Q is not an algebraic closure system in O(Y ), Theorem 4.5. The subset A(P(P ) ) ⊆A(P(P ) ,β) is a closure system with closure con con map X → X . The isomorphism A(P(P ) ,β) → N (P ),X →X maps A(P(P ) ) onto the closure system A (P ) ∩ N (P ) ⊆ N (P ), and the corre- con con sponding closure map is given by Q → Q , cf. Theorem 5.2. The set C (P ) of closure systems and the set A (P ) of algebraic closure systems in P are closure systems in P(P ), 3.1. We do not know whether N (P ) is a closure system in P(P ). However: Theorem 7.1. If P is a coherent algebraic frame then A (P ) ∩ N (P ) ⊆ P(P ) is a closure system. Proof. We abbreviate Z = P(P ) . The map A(Z ) → P(P ),Y →Y is a con P poset embedding onto A (P ) ∩ N (P ), Theorem 5.2, Corollary 6.9. Consider a subset Y⊆ A(Z ) and define X = Y. Then X is the infimum of con P {Y | Y ∈Y} in A (P ) ∩ N (P ). We have to show that X is the infimum P P of {Y | Y ∈Y} in P(P ), i.e., X = Y . The inclusion X ⊆ P P P P Y ∈Y Y holds trivially. Y ∈Y Let T ⊆ Z be patch closed and e : T → Z the inclusion. Using Proposi- tion 6.17 we identify the inclusion ι : T → P with O(e) : O(T ) →O(Z), P ∗ which sends U ∈O(T)to Z \ T \ U , the largest O ∈O(Z) with O ∩ T = U . Note that T \U is patch closed in Z, hence T \U is the set of specializa- tions of T \U.Moreover, T \U is contained in the set of specializations of min (T \ U ) , its set of minimal points, [6, Proposition 4.1.2 and Theorem 4.1.3]. Z Z Z min min min Thus T \U = (T \U ) and (T \U ) =(T \U ) . Now pick O ∈ Y , i.e., O = Z\Y \O for each Y ∈Y.The sets Y ∈Y min Y \ O coincide for all Y ∈Y, hence the sets (Y \ O) coincide as well and min are equal to (X\O) . It follows that O = Z\X\O ∈X . Proposition 7.2. Let P be a coherent algebraic frame, Z = P(P ) .Let Y ⊆ Z ↑Z ↑Y be patch closed, M = Y .For a ∈ P the sets a and a are patch closed in Z and the following conditions are equivalent: (a) a ∈ M , ↑Z min (b) (a ) ⊆ Y . Proof. We identify the closure operator η = η : P → P with the nucleus M,P ↑P ν : O(Z) →O(Z),U → Z\Y \U , Proposition 6.17.The sets a ,Y,Z ⊆ P ↑Y ↑Z are patch closed, hence a and a are patch closed as well. It follows that they are the upsets in Y and Z generated by their respective sets of minimal elements, [6, Proposition 4.1.2 and Theorem 4.1.3]. Note that a = η(a)ifand ↑Z ↑Z ↑Y only if Z\a = ν(Z\a )= Z\a . 17 Page 32 of 33 N. Schwartz Algebra Univers. (a)⇒(b). As a ∈ M , hence a = η(a), it follows from [6, Theorem 1.5.4] ↑Z ↑Y ↑Z ↑Z min ↑Y min ↑Y that a = a =(a ) , But then (a ) =(a ) ⊆ Y . ↑Z min ↑Z min ↑Y min (b)⇒(a). The inclusion (a ) ⊆ Y implies (a ) =(a ) , hence ↑Z ↑Y a = a , i.e., a = η(a) ∈ M . Corollary 7.3. Let P be a coherent algebraic frame, Z = P(P ) ,and a ∈ P . con ↑Z min Then (a ) is the smallest patch closed set X ⊆ Z with a ∈X . Declarations Funding Information Open Access funding enabled and organized by Projekt DEAL. Data Availability Statement Data sharing is not applicable to this article as no new data were created or analyzed in this study. Conflict of interest statement There are no conflicts of interest. Open Access. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and re- production in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regu- lation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons. org/licenses/by/4.0/. Publisher’s Note Springer Nature remains neutral with regard to jurisdic- tional claims in published maps and institutional affiliations. References [1] Banaschewski, B.: Coherent Frames. In: Banaschewski, B., Hoffmann, R.E. (eds.) Continuous Lattices, pp. 1–11. Lecture Notes in Math., Vol. 871, Springer, Berlin (1981) [2] Baron, S.: Note on epi in T . Can. Math. Bull. 11, 503–504 (1968) [3] Bouacida, E., Echi, O., Picavet, G., Salhi, E.: An extension theorem for sober spaces and the Goldman topology. Int. J. Math. Sci. 51, 3217–3239 (2003) [4] Burris, S., Sankanappavar, H.P.: A Course in Universal Algebra. The Millenium Edition (2012) [5] Cohn, P.M.: Universal Algebra. D. Reidel Publ. Co., Dordrecht (1981) Topology of closure systems Page 33 of 33 17 [6] Dickmann, M., Schwartz, N., Tressl, M.: Spectral Spaces. New Mathematical Monographs, No. 35, Cambridge Univ. 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Algebra Universalis – Springer Journals
Published: May 1, 2023
Keywords: Poset; Complete lattice; Algebraic lattice; Frame; Closure system; Closure operator; Spectral space; Specialization; Coarse lower topology; Scott topology; Patch topology; 06A15; 06A06; 06B35
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