Note on a subdirect representation of universal algebras

Abstract

In 1952 L. FUCHS [4] obtained an external characterization of subdirect products with two factors in case of groups, rings and Boolean algebras using Whomomorphisms (first introduced by J. H. M. WEDDERBURN [7] in 1941). 1. FL~ISCHER [3] showed in 1955 that a universal algebra of any type internally represented as subdirect product of two factor algebras (see G. BIRKHOFF [2]) c a n be constructed using Fuchs' method if and only if the relevant congruence relations are permutable. In this note we shall introduce the concept of "absolute permutability of congruence relations" which, if substituted for the pairwise permutability in Fleischer's result, yields a characterization of those families of congruences on an algebra which permit Fuchs' construction regardless of the number of factors, and which, if substituted for the pairwise permutability in Birkhoff's characterization of direct products with two factors (see [1]), yields the corresponding internal characterization of direct products with an arbitrary number of factors. A short analysis of the meaning of absolute permutability will show that only an unsatisfactorily small class of subdirect representations can be obtained from W-homomorphisms if more than two factors are involved. This essentially limits the value of W-homomorphisms f o r the purpose of an external characterization of subdirect product representations. An obvious specialization to groups and rings is given. Before proceeding to the preliminaries I want to express my thanks to Professor G. GRXTZER for helpful suggestions. We assume familiarity with such elementary concepts of the theory of universal algebras as homomorphism, direct and subdirect product and the lattice of congruence relations (see e.g. [5], Chapters 0, 3). A (universal) algebra with operations F={ .10 , . .... [~,,...}~<~ on the set A is denoted by l l = ( A ; F), its congruence lattice by ~('21), the universal relation by i, the identity relation by co, the elements in a direct product H(~II; i 6 / ) by (ai)i. Mappings stand usually behind the elements; only operations make an exception. We make free use of the following generalization of the Chinese Remainder Theorem (see [8], p. 279/280) due to G. GR;iTZER (see [5], Chapter V, exercises): We say that an algebra ~/satisfies C. R. T. (01 . . . . . 0,,) where 01, ..., 0, are a finite number of congruences on 21 if for any choice of elements a i , ..., a,, in A the relations a i ~ a.i (mod 0i(2 0s), i, ]~ I, imply the solvability of the system of congruences a i ~ x (rood Oi), iCI. C. R. T. (0l, ..., 0,,) is satisfied in ~I if and only if {01,... , 0,} are pairwise permutable and E([01, ..., 0,,]) is distributive where fs L . . . . . 0,,]) denotes the sublattice of ~(2l) generated by 0~, ..., 0,,.