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2022 | Book

Introduction Read chapter Read first chapter

A Primer of Subquasivariety Lattices

Authors: Prof. Kira Adaricheva, Dr. Jennifer Hyndman, J. B. Nation, Joy N. Nishida

Publisher: Springer International Publishing

Book Series : CMS/CAIMS Books in Mathematics

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This book addresses Birkhoff and Mal'cev's problem of describing subquasivariety lattices. The text begins by developing the basics of atomic theories and implicational theories in languages that may, or may not, contain equality. Subquasivariety lattices are represented as lattices of closed algebraic subsets of a lattice with operators, which yields new restrictions on the equaclosure operator. As an application of this new approach, it is shown that completely distributive lattices with a dually compact least element are subquasivariety lattices. The book contains many examples to illustrate these principles, as well as open problems. Ultimately this new approach gives readers a set of tools to investigate classes of lattices that can be represented as subquasivariety lattices.

Table of Contents

Frontmatter
Chapter 1. Introduction
Abstract
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) and quasivarieties (implicational classes). Then we recall the lattice theory that will be used, in particular, lattices of algebraic sets and lower bounded lattices.
Kira Adaricheva, Jennifer Hyndman, J. B. Nation, Joy N. Nishida
Chapter 2. Varieties and Quasivarieties in General Languages
Abstract
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 well-established: homomorphisms, subalgebras, direct products, ultraproducts, Galois connections, as discussed in the first chapter. It is not surprising that these tools are robust and work in a more general setting than the classical twentieth century algebra in which they arose. Thus we can develop these notions in languages that may not include equality as a predicate, and it turns out to be useful to do so.
Kira Adaricheva, Jennifer Hyndman, J. B. Nation, Joy N. Nishida
Chapter 3. Equaclosure Operators
Abstract
There is a natural closure operator Γ on the lattice \(\text{L}_{\text{q}}(\mathcal K)\) of subquasivarieties of a quasivariety \(\mathcal K\): for \(\mathcal Q \leq \mathcal K\) let \(\varGamma (\mathcal Q) = \mathcal K \cap \mathbb {HSP}(\mathcal Q) = \mathcal K \cap \mathbb {H}(\mathcal Q)\). The map Γ is called the natural equaclosure operator on \(\text{L}_{\text{q}}(\mathcal K)\). In this chapter we find 3 new properties of the natural equaclosure operator and investigate how they affect the structure of subquasivariety lattices.
Kira Adaricheva, Jennifer Hyndman, J. B. Nation, Joy N. Nishida
Chapter 4. Preclops on Finite Lattices
Abstract
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.
Kira Adaricheva, Jennifer Hyndman, J. B. Nation, Joy N. Nishida
Chapter 5. Finite Lattices as : The Case
Abstract
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 semilattice S and a set H of operators such that \(\mathbf L \cong \operatorname {Sub}(\mathbf S,\wedge ,1,H)\) with γ corresponding to the natural equaclosure operator Γ? There is an algorithm to determine this when the join irreducible elements are minimal in their γ-classes.
Kira Adaricheva, Jennifer Hyndman, J. B. Nation, Joy N. Nishida
Chapter 6. Finite Lattices as : The Case
Abstract
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.
Kira Adaricheva, Jennifer Hyndman, J. B. Nation, Joy N. Nishida
Chapter 7. The Six-Step Program: From (L, γ) to
Abstract
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 \((\text{L}_{\text{q}}(\mathcal K),\varGamma )\) for some quasivariety \(\mathcal K\) and its natural equaclosure operator Γ.
Kira Adaricheva, Jennifer Hyndman, J. B. Nation, Joy N. Nishida
Chapter 8. Lattices 1 + L as
Abstract
In this chapter we consider conditions under which, given a finite join semidistributive lattice, the linear sum \(\mathbf 1 + \mathbf L \cong \operatorname {L}_{\text{q}}(\mathcal K)\) for some quasivariety \(\mathcal K\) of structures in a language with equality. In particular, if L is isomorphic to the lattice of H-closed subsets of an algebraic lattice S with a monoid of operators H, then \(\mathbf 1 + \mathbf L \cong \operatorname {L}_{\text{q}}(\mathcal K)\) for some \(\mathcal K\) swith equality.
Kira Adaricheva, Jennifer Hyndman, J. B. Nation, Joy N. Nishida
Chapter 9. Representing Distributive Dually Algebraic Lattices
Abstract
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 \(\text{L}_{\text{q}}(\mathcal K)\) with equality. Moreover, every distributive lattice that is both algebraic and dually algebraic, and has its least element dually compact, is isomorphic to \(\text{L}_{\text{q}}(\mathcal K)\) for some quasivariety \(\mathcal K\) with equality.
Kira Adaricheva, Jennifer Hyndman, J. B. Nation, Joy N. Nishida
Chapter 10. Problems and an Advertisement
Abstract
In this chapter we propose a number of open problems about subquasivariety lattices.
Kira Adaricheva, Jennifer Hyndman, J. B. Nation, Joy N. Nishida
Backmatter
Metadata
Title
A Primer of Subquasivariety Lattices
Authors
Prof. Kira Adaricheva
Dr. Jennifer Hyndman
J. B. Nation
Joy N. Nishida
Copyright Year
2022
Electronic ISBN
978-3-030-98088-7
Print ISBN
978-3-030-98087-0
DOI
https://doi.org/10.1007/978-3-030-98088-7

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