On the definability of \(P_ S(A)\) (Q1089022)

A sufficient condition for the class, \(P_ s(A)\), of subdirect powers of a finite algebra A to be elementary was given by \textit{G. Grätzer} [Acta Sci. Math. 25, 160-168 (1964; Zbl 0192.096)]. In this paper an algebraic necessary and sufficient condition for \(P_ s(A)\) to be elementary is provided, and also a syntactic sufficient condition which is weaker than Grätzer's. (The conditions are too technical to give here.) The paper concludes by mentioning three interesting open questions.