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Varieties of complex algebras. (English) Zbl 0722.08005

Any \(n+1\)-ary relation R on a set X induces the n-ary operation \(f_ R\) on the set \({\mathcal P}(X)\) of subsets of X defined by \[ f_ R(X_ 1,...,X_ n)=\{y| \exists x_ 1...x_ n(R(x_ 1,...,x_ n,y)\wedge x_ 1\in X_ 1\wedge...\wedge x_ n\in X_ n)\}. \] For any relational structure \(S=<X;R>\), any subalgebra of the algebra \(S^+=<{\mathcal P}(X);\{f_ R| R\in {\mathcal R}\}>\) is called a complex algebra. For any class K of relational structures, \(K^+\) denotes the closure under isomorphism of \(\{S^+| S\in K\}\). \(SK^+\) is often a variety, and this is frequently associated with K being elementary. Examples include the varieties of closure algebras, relation algebras, cylindric algebras, modal algebras, and numerous of subvarieties of these. The aim of the paper is “to study the general situation here, focusing on these related occurrences of definability: elementary K determining equational \(SK^+\)”.
Reviewer’s remark: A. A. Makhmudov [Mathematical conference dedicated to the memory of M. Souslin, Saratov, 1991] proves the following: 1) \(U_ pK^*\subseteq S(U_ pK)^*\) for any class K, where \(K^*=\{A^*| A\in K\}\) and \(A^*\) is obtained from \(A^+\) by removing \(\emptyset\) (and the Boolean operations); 2) \(ISK^*\) is a universal class for any universal class K; 3) \(ISK^*\) is a quasivariety for any class K closed under reduced products; 4) \(HSK^*\) is a variety for any \(K=P(K)\).

MSC:

08B99 Varieties
03C05 Equational classes, universal algebra in model theory
03C40 Interpolation, preservation, definability
03C60 Model-theoretic algebra
08A02 Relational systems, laws of composition
03G15 Cylindric and polyadic algebras; relation algebras
03G25 Other algebras related to logic
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