Operations with respect to which the elements of a Boolean algebra form a group

Abstract

In a. previous papert I pointed out the existence of two operations with respect to each of which the elements of a boolean algebra form an abelian group. If we denote the logical sum of two elements a, b by a + b, theii logical product by a b, and the negative of an element a by a', then the two operations in question are given by ab' + a'b, ab + a'b'. In the present paper I determine all the operations with respect to which the elements of a boolean algebra form a group in general and an abelian group in particular. Postulates for groups.t A class K of elements a, b, c, . . . is a group with respect to an operation 0 if the following two conditions are satisfied: P1. (aOb)Oc = aO (bOc), whenever a, b, c, a0b, b0c, aO(bOc) are elements of K. P2. For any two elements a, b, in K there exists an element x such that aox = b. The group is abelian if the following condition also is satisfied: P3. aob = bOa, whenever a, b, b Oa are elements of K. Determination of group operations. We shall have all the operations of a boolean algebra with respect to which the elements form a group if we determine for groups in general all the boolean operations which have the properties P1, P2, and for abelian groups, all the operations which have the properties P1, P2, Ps. I proceed to effect this determination. If f(x, y) is any determinate function of two elements x, y of a boolean algebra, then f(X, y) = f(1, 1)xy+f(1, O)xy'+f(O, 1)x'y+f(O, O)x'y',