Acta Mathematiea Academiae Scientiarum Hungaricae Tomus 38 (1---4), (1981), 9~14. DIRECT AND SUBDIRECT DECOMPOSITIONS OF UNIVERSAL ALGEBRAS WITH A BOOLEAN ORTHOGONALITY By W. H. CORNISH (Bedford Park) and P. N. STEWART* (Halifax) 1. Introduction. In this paper decompositions of universal algebras with a Boolean orthogonality as subdirect, strongly irredundant subdirect and direct products are obtained. In particular, we prove the direct product theorem on which many of the results in [6] depend. The prototypical example of an algebra with a Boo- lean orthogonality is an associative ring with no proper nilpotent elements: the rela- tion _k defined by x _l_y if and only if xy=O is a Boolean orthogonality and the fac- tors in our subdirect decompositions are rings with no proper divisors of zero. We refer to GRs book [8] for the conventions and fundamental concepts of universal algebra. In particular, the set of elements of an algebra, which is denoted by a capital Gothic letter, will be denoted by the corresponding Latin letter. Whene- ver more than one algebra is under discussion it is assumed that all the algebras are of the same type. The direct product of a set of algebras {91~: 2EA} is denoted by //914, and eu denotes the projection of//91x onto 91u. It will be convenient to always regard a subdirect product as a subalgebra of the direct product. Ever5, algebra is assumed to be finitary and to possess a distinguished element 0 which is a nultary operation. Let 91 be an algebra. If 0 is a congruence on 91, then ker O={aEA: a~0(0)} is the kernel of 0. A subset J of A is an ideal if it is the kernel of some congruence on 92. If J is an ideal, then 0 (or) denotes the smallest congruence which has Y as its kernel. If 91 and 91~ are algebras and f:91 ~ 2[1 is a homomorphism, then kerf= {a EA :f(a)= 0}. Notice that kerfis an ideal. The general theory of Boolean orthogonalities has been developed by the first author in [5]. For the convenience of the reader we give the basic definitions as applied to our situation. Let _t_ be a binary relation on the set A of elements of an algebra 91. For X~ A, Xa={aEA:a• for all xEX}; X•177 denotes (X•177 and x • denotes {x}• for each x EA. The set N (91)= {Y: Y= X • for some X= A} is called the set of potars. With this notation a binary relation j_ is a Boolean orthogonality on 91 if and only if the follow- ing are satisfied: (A1) x• (Ay) x • y implies y _i_ x, (Aa) x • x implies x = 0, (As) x •177 ~y•177 ={0} implies xly, for all x, yEA, and (An) every polar is an ideal. * The research of the second named author was supported by National Research Council Grant A--8789. Acta Mathema~ica Acade77z~ae Scientiarum Hungaricae 38, 19.9l