Pair-dense Relation Algebras

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 fha; ai ; hb; big (with a = b allowed). In a simple pair-dense relation algebra, every pair is either a \point" (an algebraic analogue of fha; aig) or a \twin" (a pair which contains no point). In fact, every simple pair-dense relation algebra A is completely representable over a set U ii jUj = + 2, where is the number of points of A and is the number of twins of A. 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 A is completely representable over U ii jUj = , where is the number of points of A. 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 x 0 ; : : : ;x ?1 , in case one of the x 's is 1. This result is proved in a general setting for a special class of groupoids. x1. Introduction In its most basic form, a representation result for relation algebras is simply a theorem which asserts that every relation algebra having a certain property is repre-sentable. This paper presents several new theorems of this kind. The importance of such results stems from the fact that not all relation algebras are representable. Until Lyndon's counterexample in L50], one could have hoped (as in T41], pp. 87{88) that the ultimate representation result would be true, namely that every relation algebra would be representable. This happy situation exists for groups and Boolean