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## Über dieses Buch

This book discusses the ways in which the algebras in a locally finite quasivariety determine its lattice of subquasivarieties. The book starts with a clear and comprehensive presentation of the basic structure theory of quasivariety lattices, and then develops new methods and algorithms for their analysis. Particular attention is paid to the role of quasicritical algebras. The methods are illustrated by applying them to quasivarieties of abelian groups, modular lattices, unary algebras and pure relational structures. An appendix gives an overview of the theory of quasivarieties. Extensive references to the literature are provided throughout.

## Inhaltsverzeichnis

### Chapter 1. Introduction and Background

The properties of a quasivariety are reflected in the structure of its lattice of subquasivarieties. For example, a subquasivariety $$\mathcal{S}$$ of a quasivariety $$\mathcal{K}$$ is finitely based relative to $$\mathcal{K}$$ if and only if $$\mathcal{S}$$ is dually compact in the lattice of subquasivarieties of $$\mathcal{K}$$. In order to understand how quasivarieties work, we need general methods to analyze their lattices of subquasivarieties.
Jennifer Hyndman, J. B. Nation

### Chapter 2. Structure of Lattices of Subquasivarieties

Remember that the lattice $$\mathop{\mathrm{L_{q}}}\nolimits (\mathcal{K})$$ of all subquasivarieties of a quasivariety $$\mathcal{K}$$ is dually algebraic and join semidistributive. The goal of this section is to characterize the completely join irreducible quasivarieties in $$\mathop{\mathrm{L_{q}}}\nolimits (\mathcal{K})$$. Most of the results in this section can be found in Section 5.​1 of Gorbunov’s book [77].
Jennifer Hyndman, J. B. Nation

### Chapter 3. Omission and Bases for Quasivarieties

In this section we show that if $$\mathcal{K}$$ is a locally finite quasivariety of finite type, T is a finite algebra in $$\mathcal{K}$$, and αΔ in $$\mathop{\mathrm{Con}}\nolimits _{\mathcal{K}}\,\mathbf{T}$$, then the quasivariety $$\langle \varepsilon _{\mathbf{T},\alpha }\rangle$$ consists of all algebras in $$\mathcal{K}$$ that omit a finite list of forbidden subalgebras. A slight variation finds the quasivarieties that are minimal with respect to not being contained in $$\langle \varepsilon _{\mathbf{T},\alpha }\rangle$$. Both these results are in Theorem 3.4. As a consequence, subquasivarieties that are finitely based relative to $$\mathcal{K}$$ can be characterized by the exclusion of finitely many subalgebras (Theorem 3.10).
Jennifer Hyndman, J. B. Nation

### Chapter 4. Analyzing

To further investigate the structure of $$\mathop{\mathrm{L_{q}}}\nolimits (\mathcal{K})$$, where $$\mathcal{K}$$ is a locally finite quasivariety of finite type, we want algorithms to determine
(1)
the quasicritical algebras T in $$\mathcal{K}$$,

(2)
the order on join irreducible quasivarieties, i.e., when $$\langle \mathbf{T}\rangle \leq \langle \mathbf{S}\rangle$$,

(3)
the join dependencies, i.e., when $$\langle \mathbf{T}\rangle \leq \langle \mathbf{S}_{1}\rangle \vee \cdots \vee \langle \mathbf{S}_{n}\rangle$$ nontrivially.

Jennifer Hyndman, J. B. Nation

### Chapter 5. Unary Algebras with 2-Element Range

The previous sections have included various algorithms for working with locally finite quasivarieties of finite type. We will illustrate these algorithms by applying them to quasivarieties contained in the variety $$\mathcal{M}$$ generated by a particular 3-element algebra M described below.
Jennifer Hyndman, J. B. Nation

### Chapter 6. 1-Unary Algebras

We are particularly interested in the proper subvarieties of $$\mathcal{N}$$ and $$\mathcal{N}^{0}$$ that are determined by the equation $$f^{r}x \approx f^{s}x$$ for some pair r < s. Thus, for 0 ≤ r < s, we consider the following varieties.
Jennifer Hyndman, J. B. Nation

### Chapter 7. Pure Unary Relational Structures

The methods developed for quasivarieties in the preceding sections apply to systems more general than just algebras. For something a little different, let us turn our attention to pure unary relational structures, which are sets with finitely many unary predicates A 1, , A k.
Jennifer Hyndman, J. B. Nation

### Chapter 8. Problems

Much is known about specific classes of algebras. If T is a finite quasiprimal algebra, then $$\mathop{\mathrm{L_{q}}}\nolimits (\langle \mathbf{T}\rangle )$$ is finite; see Section A.6, also Blanco et al. [34]. The atoms of $$\mathop{\mathrm{L_{q}}}\nolimits (\mathcal{K})$$ were discussed in Section 4.​4. Every 2-element algebra generates an atom, but Adams and Dziobiak [2] exhibit a 3-element algebra that generates a Q-universal quasivariety. Sapir showed that there is a Q-universal quasivariety generated by a finite commutative 3-nilpotent semigroup [152].
Jennifer Hyndman, J. B. Nation

### Backmatter

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