The Class-Ring in Multiplicative Systems

Abstract

NOTATION. z denotes a system closed to a binary operation; a, b, c, , are elements of E-subsets A, B, C, of 1; u, v, w, * , are elements of X. The term E-set implies that the classes Cw, = w A are disjoint, the relation between the elements in Ci,, being an equivalence relation RA . Hence if w be any element in A, a, , a2 any elements in A then there exists an element a such that w = w -a; w = (w-al)-a2; (w.al).a2 = wa. These classes will be regarded as the elements of a multiplicative system T.4 in which multiplication will be denoted by X. In group theory X may be derived from *, e.g. a X b may denote any one of a *b, b *a, a * b a-l b l, b-1 * a * b etc. On the other hand X may be defined without reference to * as in the theory of rings of systems closed to two operations, e.g. ab ba, ab + ba.1 In order to establish an automorphism between the structure (lattice) of equivalence relations RA, in which A ranges over all E-subsets of A, and those quotients of E-sets having the same RA , we shall assume: CANCELLATION LAW I. If for any element u we have u a = u -b then there exists a d in D = A n B such that u a = u-b = u d, i.e. we assume that D is non-void. ASSOCIATIVE LAWS. w* (a b) = (w a).b; (w a).b = w. (a b) where b is in B.. Hence A * (B * C) = (A * B) * C. The product w-ai X v ai is in z and so falls into one and only one class CU = u A. In ordinary algebra several classes arise, viz.: (a) All elements w . A X v . A are in the same class (w X v) A. This holds in the theory of normal co-set expansions in groups and semi-groups. (b) All elements w A X v A are in the same class u A where u 5 w X v. This case cannot occur here for since A is an E-set, right units a. , a, exist such that w = w a. and v = v a,. Hence w X v is in w A X v * A which product must therefore be in (w X v) A. (c) The elements w -A X v A fall into different classes, e.g. the classes of conjugate elements in group theory.