Extending Congruence Relations

If 91 and 23 are algebras, where 9Is23, and 6 is a congruence relation on 91, let 0® be the smallest congruence relation on 93 containing 0. 91 is called a congruence subalgebra of 93 if 9ls93 and, for every congruence relation 0 on 91, 0*n|9I|2=0. Elementary subalgebras are congruence subalgebras, and there are Directed Union and Loewenheim-Skolem Theorems for congruence subalgebras analogous to those for elementary subalgebras. Consequently we obtain full analogues of the Jonsson-Morley-Vaught results concerning homogeneous-universal algebras, where the notion of "subalgebra" is everywhere replaced by "congruence subalgebra". In this note we shall consider algebras <&=(A, Q,)iei of similarity type teto1 and the corresponding first order language Lt. Our notation and terminology follow a widely accepted pattern, and details can be found in Grätzer [4] and Morley and Vaught [9]. If A'is a set then k(X) is the power or cardinality of X. 91<93 means that 91 is an elementary subsystem of 93. For later reference we state a version of the familiar "Robinson Lemmas" (see Robinson [10]). Lemma 1. If 91 < 93 and, for each keK, Rk is a binary relation over |9I|, then there exist QC and binary relations Rk over \(£\, for each keK, such that 93<£ and (91, Rk%EK<(£, R^)keK. If 91= 93 and 0 is a congruence relation on 91, we denote by 0® the smallest congruence relation on 93 containing 0. A characterization of 0s can be found in Grätzer [4, p. 55]. 91 is called a congruence subalgebra of 93 (in symbols 91<]93) if 91=93 and, for every congruence relation 0 on 91, 0®n|91|2=0. We illustrate this notion with a few remarks. It is obvious that for Abelian groups 91s 93 if and only if 9I<93. The same is true for distributive lattices and for Boolean algebras, as is well known. Congruence subalgebras play an essential role in the embedding theorems of Higman, Neumann Received by the editors April 8, 1970 and, in revised form, March 31, 1971. AMS 1969 subject classifications. Primary 5030.