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 u is completely representable over a set U iff |U| = k + 2λ, where k is the number of points of u and λ is the number of twins of u. 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 u is completely representable over U iff |U| = k, where k is the number of points of 21. 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; …; xα-1, in case one of the xk’s is 1. This result is proved in a general setting for a special class of groupoids. © 1991 American Mathematical Society.