Model theoretic properties in the variety generated by a primal algebra
type: Item
instance of: scholarly article
title: Model theoretic properties in the variety generated by a primal algebra
zbMATH Open document ID: 0795.03040
DOI: 10.1007/BF01195381
published in: Algebra Universalis
publication date: 1994-09-01T00:00:00Z
review text: Let \({\mathbf P}\) denote a primal algebra (i.e. a finite algebra for which every function is a term function), let \({\mathbf B}\) be a Boolean algebra and let \({\mathbf P}[{\mathbf B}]\) denote the bounded Boolean power of \({\mathbf P}\) by \({\mathbf B}\). Let \(\text{Th}({\mathbf A})\) denote the equational theory of an algebra \({\mathbf A}\). Then we have: Proposition. Let \({\mathbf P}\) be a primal algebra in a finite language; then \(\text{Th}({\mathbf P}[{\mathbf B}])\) is finitely axiomatizable if and only if \(\text{Th}({\mathbf B})\) is finitely axiomatizable. The paper contains many results on model completeness which require too many technical definitions easily to be stated here. However, an interesting question posed at the end of the paper which can easily be stated is: Question: What structures \({\mathbf A}\) of finite type have the property that \(\text{Th}({\mathbf A}[{\mathbf B}])\) is finitely axiomatizable if and only if \(\text{Th}({\mathbf B})\) is finitely axiomatizable?
Mathematics Subject Classification ID: 03C05
Mathematics Subject Classification ID: 08A40
Mathematics Subject Classification ID: 08B99
zbMATH DE Number: 488266
zbMATH Keywords: finite axiomatizability
zbMATH Keywords: primal algebra
zbMATH Keywords: bounded Boolean power
zbMATH Keywords: equational theory
zbMATH Keywords: model completeness
author: George C. Nelson
reviewed by: Sheila Oates-Williams
MaRDI profile type: Publication
cites work: Boolean powers
cites work: Bounded Boolean powers and \(\equiv_n\)
cites work: Model companions for finitely generated universal Horn classes
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cites work: Recursive prime models for Boolean algebras
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description: scientific article; zbMATH DE number 488266