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Transactions of the American Mathematical Society

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Finite basis theorems for relatively congruence-distributive quasivarieties


Author: Don Pigozzi
Journal: Trans. Amer. Math. Soc. 310 (1988), 499-533
MSC: Primary 08C15; Secondary 03B05, 03C05, 68Q65
DOI: https://doi.org/10.1090/S0002-9947-1988-0946222-1
MathSciNet review: 946222
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Abstract: $ \mathcal{Q}$ is any quasivariety. A congruence relation $ \Theta $ on a member $ {\mathbf{A}}$ of $ \mathcal{Q}$ is a $ \mathcal{Q}$-congruence if $ {\mathbf{A}}/\Theta \in \mathcal{Q}$. The set $ Co{n_\mathcal{Q}}{\mathbf{A}}$ of all $ \mathcal{Q}$-congruences is closed under arbitrary intersection and hence forms a complete lattice $ {\mathbf{Co}}{{\mathbf{n}}_\mathcal{Q}}{\mathbf{A}}$. $ \mathcal{Q}$ is relatively congruence-distributive if $ {\mathbf{Co}}{{\mathbf{n}}_\mathcal{Q}}{\mathbf{A}}$ is distributive for every $ {\mathbf{A}} \in \mathcal{Q}$. Relatively congruence-distributive quasivarieties occur naturally in the theory of abstract data types. $ \mathcal{Q}$ is finitely generated if it is generated by a finite set of finite algebras. The following generalization of Baker's finite basis theorem is proved. Theorem I. Every finitely generated and relatively congruence-distributive quasivariety is finitely based. A subquasivariety $ \mathcal{R}$ of an arbitrary quasivariety $ \mathcal{Q}$ is called a relative subvariety of $ \mathcal{Q}$ if it is of the form $ \mathcal{V} \cap \mathcal{Q}$ for some variety $ \mathcal{V}$, i.e., a base for $ \mathcal{R}$ can be obtained by adjoining only identities to a base for $ \mathcal{Q}$. Theorem II. Every finitely generated relative subvariety of a relatively congruence-distributive quasivariety is finitely based. The quasivariety of generalized equality-test algebras is defined and the structure of its members studied. This gives rise to a finite algebra whose quasi-identities are finitely based while its identities are not. Connections with logic and the algebraic theory of data types are discussed.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1988-0946222-1
Keywords: Quasivariety, finite basis theorem, Baker's theorem, relatively congruence-distributive, abstract data type
Article copyright: © Copyright 1988 American Mathematical Society