On taking subalgebras of relativized relation algebras (Q1105597)

Let \({\mathfrak A}=<A,(f)_{f\in F}>\) be an algebra and \(\cdot\) a distinguished binary operation of \({\mathfrak A}\). For each \(a\in A\) set \({\mathfrak Rl}_ a{\mathfrak A}=<Rl_ a{\mathfrak A},(f')_{f\in F}>\), where \(Rl_ a{\mathfrak A}=\{a\cdot x:\) \(x\in A\}\) and \(f'(x_ 1,...,x_ n)=a\cdot f(x_ 1,...,x_ n)\) for every \(f\in F\) and \(x_ 1,...,x_ n\in Rl_ a{\mathfrak A}\). Let RA be the class of all relation algebras and RRA the class of all representable relation algebras. For any class K of similar algebras set \(Rl'K=\{{\mathfrak Rl}_ a{\mathfrak A}:\) \({\mathfrak A}\in K\), \(a\in A\}\) and \(SK=\{{\mathfrak B}:\) \({\mathfrak B}\subseteq {\mathfrak A}\) for some \({\mathfrak A}\in K\}\). The author proves that for any class K satisfying \(RRA \subseteq K\subseteq RA\) the following properties hold: (i) \(Rl'K \neq SRl'K\); (ii) there is a complete and atomic member of SRl'K-Rl'K; (iii) there is a complete and atomic member of SRl'RRA-Rl'RA. The relationship to similar problems in the theory of cylindric algebras is discussed.