Baldwin, John T.; Samhan, Marouf A. On the definability of \(P_ S(A)\). (English) Zbl 0618.08005 Algebra Univers. 23, 254-261 (1986). A sufficient condition for the class, \(P_ s(A)\), of subdirect powers of a finite algebra A to be elementary was given by 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. Reviewer: S.Oates-Williams Cited in 1 Document MSC: 08C10 Axiomatic model classes 03C60 Model-theoretic algebra Keywords:first order sentences; generalized cryptoelementary definability; subdirect powers of a finite algebra Citations:Zbl 0192.096 PDFBibTeX XMLCite \textit{J. T. Baldwin} and \textit{M. A. Samhan}, Algebra Univers. 23, 254--261 (1986; Zbl 0618.08005) Full Text: DOI References: [1] G. Gr?tzer,On the class of subdirect powers of a finite algebra, Acta. Sci. Math.,25 (1964), 160-168. · Zbl 0192.09601 [2] G.Gr?tzer,Universal Algebra, Springer-Verlag, 1979. [3] K. Hickin, J. Plotkin,On the equivalence of three local theorem techniques, Proceedings of the A.M.S.,35 (1972), 389-391. · Zbl 0258.20038 · doi:10.1090/S0002-9939-1972-0302765-9 [4] D. McLain,Local theorems in universal algebras, J. London Math. Soc.34 (1959), 177-184. · Zbl 0085.01902 · doi:10.1112/jlms/s1-34.2.177 [5] M. Samhan,First order definability in universal algebra, Ph.D. thesis, University of Illinois at Chicago, 1985. [6] A.Tarski,Contributions to the theory of models I, II, III, Nederl. Akad. Wetensch. Proc. Ser. A. 57/58=Indag. Math. 16/17 (1954/55) 572-581, 582-588, 56-64. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.