Strong congruences and Mal'cev conditions on first order structures
by
Marcel Tonga
University of Yaounde-1(Cameroon); Fac. of Science; Dept. of Mathematics

The notions of *-congruence and *-variery were introduced in [W]. If A = (A, R, F) is a first order structure and \theta in Con(A), then \theta is a *-congruence if given any r in R, m-ary, and [( --> ) || a], [( --> ) || b] in A^m, such that á a_i, b_i ñ in \theta, 1 <= i <= n, then [( --> ) || a] in r iff [( --> ) || b] in r. We denote the set of *-congruences of A by Con_*(A). A *-morphism is a morphism whose kernel is a *-congruence. A class K of structures closed under substructure, product and *-image is called a *-variety. In [W], Mal'cev conditions are established for *-varieties.

Using the construction of [W], we show that for any structure A = (A, R, F) with F =/= \phi, there are structures B in H^*SP(frakA) with non trivial *-congruences. We characterize structures whose *-congruence compatible functions are interpolated on classes of *-congruence by term functions. Finally, we note that if the discriminator function is interpolated by term functions on *-congruence classes, then Con_*(A) is arithmetical.

[W] N. Weaver. Generalized varieties, Algebra Universalis, vol.30 (1993), 27-52.

Date received: May 19, 2000