Model theoretic properties in the variety generated by a primal algebra
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?
scientific article; zbMATH DE number 488266
scientific article