The number of \(Q\)-congruences in positive algebras (Q1317611)

Effective description of positive models in various classes is one of the basic problems in abstract data type theory [\textit{S. S. Goncharov}, Vychisl. Sist. 107, 52-70 (1985; Zbl 0621.68020)]. Aspects of definability of positive algebras by universal Horn sentences in classes of their effective homomorphic images play an important role in logic programming [\textit{J. W. Lloyd}, Foundations of logic programming (1984; Zbl 0547.68005)]. This range of problems turned out to be closely connected with decidability problems for positive enumerations of these algebras. For example, every recursively enumerable set of universal Horn sentences that is true in a nonrecursive positive algebra is also true in some positive quotient algebra of this algebra [the author, Vychisl. Sist. 133, 3-13 (1990; Zbl 0752.03016)]. In what follows we prove that every nonrecursive positive algebra, which is completely defined in the class of its recursive quotient algebras by a recursively enumerable set \(Q\) of universal Horn sentences, possesses a continuum of congruences such that \(Q\) is also true in quotient algebras modulo these congruences. We also demonstrate a number of applications of this fact.