Axiomatizing complex algebras by games. (Q1771885)

Given an algebra \({\mathcal A}=(A,F)\), a power algebra is the power set \(P(A)\) equipped with operations \(F^{\circ }\) defined by lifting: if \(f \in F\) is \(n\)-ary and \(X_i \subseteq A\) for \(i=1,\dots ,n\) then \(f^{\circ }(X_1,\dots ,X_n)=\{f(x_1,\dots ,x_n);\;x_i \in X_i\}\). A full complex algebra is then an algebra \({\mathcal A}^{+}=(P(A),\cup ,-,\varnothing ,F^{\circ })\). Main theorem: Let \(\mathcal V\) be a variety of a finite type \(\Sigma \). There is a set \(\Phi (\mathcal V)\) of universal first-order sentences in the language of complex algebras over \({\mathcal V}\) such that whenever \({\mathcal A}\) is a Boolean algebra with \(\Sigma\)-operators, \(\mathcal A\models \Phi (\mathcal V)\) if and only if \(\mathcal A\) is representable as a complex algebra over \(\mathcal V\). \(\Phi (\mathcal V)\) can be obtained effectively from the axiomatization of \(\mathcal V\), and if this axiomatization is recursively enumerable then \(\Phi (\mathcal V)\) is recursive.