On algebras whose factor algebras are Boolean
J. M. G. Fell, Alfred Tarski · Pacific Journal of Mathematics · 1952
Introduction. 1 We consider an algebraic system 2Ϊ = {A, +) constituted by an arbitrary set A and a binary operation +.The set A is assumed to be closed under +, and to contain a (uniquely determined) zero element, that is, an element 0 such that for every x in A. We shall refer to such a system simply as an algebra.By a subalgebra B of ?I we understand an arbitrary subset of A which is closed under + and contains 0 as an element.For two subalgebras B and C, a function /, whose domain includes (but does not necessarily coincide with) B and which maps B onto a subset of C in such a way that for all b l9 b 2 in B, is called as usual a (B, C) -homomorphism; if in addition /is biunique on B, and maps B onto the whole of C, it is called a (B, C)-isomorphism.A relation R holding between certain pairs of elements of A is called a congruence relation over 21 if (i) R is an equivalence relation whose field is A, and (ii) (α + b) /?(α' + b') whenever α/?α'and bRb'; here we have expressed symbolically by cRd the statement that R holds between the elements c and d.Suppose that R is a congruence relation over 21.For each element a of A, the coset of a under R, in symbols a/R, will be defined as the set of all b in A for which aRb.It is easy to see that any two cosets are either identical or else have no element in common, and that the set-theoretical union of all the cosets is simply A. If a/R and b/R are any two cosets, we denote by a/R +'b/R the coset (α + b)/R.By condition (ii) of the definition of a congruence relation, this determines an operation +' on pairs of cosets.The algebra consisting of *A detailed discussion of all the notions and results contained in the Introduction will be found in [8] (see in particular Appendix § § A and B) and [4] (see in particular § §1 and 2).