Subdirect Unions in Universal Algebra

The number of distinct operations (that is, the range of the variable a) may be infinite, but for our main result (Theorem 2), we shall require every n(a) to be finite—that is, it will concern algebras with finitary operations. The concepts of subalgebra, congruence relation on an algebra, homomorphism of one algebra A onto (or into) another algebra with the same operations, and of the direct union AiX • • • XAr of any finite or infinite class of algebras with the same operations have been developed elsewhere. More or less trivial arguments establish a manyone correspondence between the congruence relations 0i on an algebra A and the homomorphic images Hi — 6i(A) of the algebra (isomorphic images being identified); moreover the congruence relations on A form a lattice (the structure lattice of A). In this lattice, the equality relation will be denoted 0 ; all other congruence relations will be called proper. More or less trivial arguments also show (cf. Lattice theory, Theorem 3.20) that the isomorphic representations of any algebra A as a subdirect union, or subalgebra S^HiX • • • XHr of a direct union of algebras Hi, correspond essentially one-one to the sets of congruence relations 0i on A such that A0t=O. In fact, given such a set of Oi, the correspondence