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A Primer of Subquasivariety Lattices

Part of the CMS/CAIMS Books in Mathematics Book Series (volume 3)

DOI: 10.1007/978-3-030-98088-7

ISBN: 978-3-030-98087-0

Publisher: Springer International Publishing - Springer Journals

May 10, 2022

1 - 10 of 10 Chapters

LitStream

Chapter 1Introduction

Adaricheva, Kira; Hyndman, Jennifer; Nation, J. B.; Nishida, Joy N.

Pages: 1 –25

[This monograph studies the lattice of subquasivarieties of a quasivariety. We consider both the properties that these lattices have, and which complete lattices can be represented as lattices of subquasivarieties. The first chapter reviews some classical results on varieties (equational classes)...
LitStream

Chapter 2Varieties and Quasivarieties in General Languages

Adaricheva, Kira; Hyndman, Jennifer; Nation, J. B.; Nishida, Joy N.

Pages: 27 –68

[In this chapter we will develop the fundamental notions of varieties and quasivarieties, starting near the beginning (just past set theory). The key idea is to link logical theories (equational theories or implicational theories) with models (varieties or quasivarieties). The tools are...
LitStream

Chapter 3Equaclosure Operators

Adaricheva, Kira; Hyndman, Jennifer; Nation, J. B.; Nishida, Joy N.

Pages: 69 –91

[There is a natural closure operator Γ on the lattice Lq(K)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek}...
LitStream

Chapter 4Preclops on Finite Lattices

Adaricheva, Kira; Hyndman, Jennifer; Nation, J. B.; Nishida, Joy N.

Pages: 93 –124

[A preclop is a closure operator on a lattice that satisfies some of the known properties of the equational closure operator on subquasivariety lattices. In this chapter we look at preclops on finite lattices and give an algorithm to decide if a finite lattice supports a preclop.]
LitStream

Chapter 5Finite Lattices as Sub(S,∧,1,H)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\operatorname {Sub}(\mathbf S,\wedge ,1,H)$$ \end{document}: The Case J(L)⊆τ(L)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\operatorname {J}(\mathbf L) \subseteq \tau (\mathbf L)$$ \end{document}

Adaricheva, Kira; Hyndman, Jennifer; Nation, J. B.; Nishida, Joy N.

Pages: 125 –138

[In this chapter and the next, we address the question: Given a pair (L, γ) consisting of a finite lower bounded lattice and a preclop, when can we find a finite semilatticeSand a setHof operators such thatL≅Sub(S,∧,1,H)\documentclass[12pt]{minimal} \usepackage{amsmath}...
LitStream

Chapter 6Finite Lattices as Sub(S,∧,1,H)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\operatorname {Sub}(\mathbf S,\wedge ,1,H)$$ \end{document}: The Case J(L)⊈τ(L)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\operatorname {J}(\mathbf L) \not \subseteq \tau (\mathbf L)$$ \end{document}

Adaricheva, Kira; Hyndman, Jennifer; Nation, J. B.; Nishida, Joy N.

Pages: 139 –159

[In this chapter, we turn to the case of representing a finite lattice by subsemilattices with operators when the join irreducibles are not least in their γ-classes. There is no algorithm for this case, but there are reasonably systematic and effective ad hoc methods.]
LitStream

Chapter 7The Six-Step Program: From (L, γ) to (Lq(K),Γ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$(\text{L}_{\text{q}}(\mathcal K),\varGamma )$$ \end{document}

Adaricheva, Kira; Hyndman, Jennifer; Nation, J. B.; Nishida, Joy N.

Pages: 161 –216

[In this chapter, we concentrate on a method for representing pairs (L, γ), where L is a finite lower bounded lattice and γ an equaclosure operator on it, as (Lq(K),Γ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts}...
LitStream

Chapter 8Lattices 1 + L as Lq(K)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\operatorname {L}_{\text{q}}(\mathcal K)$$ \end{document}

Adaricheva, Kira; Hyndman, Jennifer; Nation, J. B.; Nishida, Joy N.

Pages: 217 –236

[In this chapter we consider conditions under which, given a finite join semidistributive lattice, the linear sum 1+L≅Lq(K)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy}...
LitStream

Chapter 9Representing Distributive Dually Algebraic Lattices

Adaricheva, Kira; Hyndman, Jennifer; Nation, J. B.; Nishida, Joy N.

Pages: 237 –253

[We show that every distributive dually algebraic lattice can be represented as Sp(S, H) with S an algebraic lattice and H a monoid of operators. As a consequence, every linear sum 1 + D with D distributive and dually algebraic is isomorphic to a lattice of subquasivarieties...
LitStream

Chapter 10Problems and an Advertisement

Adaricheva, Kira; Hyndman, Jennifer; Nation, J. B.; Nishida, Joy N.

Pages: 255 –259

[In this chapter we propose a number of open problems about subquasivariety lattices.]

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