Pair-dense relation algebras

Abstract

The central result of this paper is that every pair-dense relation algebra is completely representable. A relation algebra is said to be pair-dense if every nonzero element below the identity contains a pair. A pair is the relation algebraic analogue of a relation of the form {(a, a), (b, b)} (with a = b allowed). In a simple pair-dense relation algebra, every pair is either a point (an algebraic analogue of {(a, a)}) or a twin (a pair which contains no point). In fact, every simple pair-dense relation algebra 2( is completely representable over a set U iS | Ul = K + 2R, where K iS the number of points of 2( and A is the number of twins of 2( . A relation algebra is said to be point-dense if every nonzero element below the identity contains a point. In a point-dense relation algebra every pair is a point, so a simple point-dense relation algebra 2( is completely representable over U iS IUI = c, where K iS the number of points of 2( . This last result actually holds for semiassociative relation algebras, a class of algebras strictly containing the class of relation algebras. It follows that the relation algebra of all binary relations on a set U may be characterized as a simple complete point-dense semiassociative relation algebra whose set of points has the same cardinality as U . Semiassociative relation algebras may not be associative, so the equation (x; y); z = x; (y; z) may fail, but it does hold if any one of x, y, or z is 1. In fact, any rearrangement of parentheses is possible in a term of the form x0; ...; X(X_1 X in case one of the XKSS is 1. This result is proved in a general setting for a special class of groupoids.