A Representation for Boolean Algebras

Abstract

1. The content. The success of Stone [5] in representing Boolean algebras by fields of sets leads one to search for other representations. (See also [2, p. 159], [4].) Using splitting endomorphisms on a type of Abelian group with an order relation, we are led to a faithful representation of a given Boolean algebra B. The group employed will be called a Boolean group. A special case of a Boolean group is a Boolean ring, and we shall show that 8 can be represented by a set of endomorphisms on its corresponding Boolean ring. If ? is complete the group on which it operates will be lower complete and upper conditionally complete. The Boolean groups are themselves special cases of a wider class of groups with an order relation, the vector ordered groups. Vector ordered groups generalize direct sums of Abelian groups, and their splitting endomorphisms form Boolean algebras. A simple example illustrates virtually the entire theory: Let ZJ be the ring of integers modulo 2, and let (M be the Cartesian product of some infinite collection of copies of 2*. 5 can be made into the strong direct sum of the 32 by introducing component-wise addition and multiplication. Then @ becomes a Boolean ring with a unity. It is also a Boolean algebra [2, p. 154] if one writes x ? y whenever each component of y ? (5 equals the corresponding component of x ? (M or is zero. Let 'P (() be the set of all endomorphisms V of the group structure of (M for which 12 _ v and V (x) ? x for every x P (M. If an order relation is defined in 'P in the obvious way, 'P becomes a Boolean algebra isomorphic to the Boolean algebra 5, so that the algebra 05 is repre