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 $\{ \langle {a,a} \rangle ,\langle {b,b} \rangle \}$ (with $a= b$ allowed). In a simple pair-dense relation algebra, every pair is either a "point" (an algebraic analogue of $\{ \langle {a,a} \rangle \}$) or a "twin" (a pair which contains no point). In fact, every simple pair-dense relation algebra $\mathfrak {A}$ is completely representable over a set $U$ iff $|U|= \kappa + 2\lambda$, where $\kappa$ is the number of points of $\mathfrak {A}$ and $\lambda$ is the number of twins of $\mathfrak {A}$. 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 $\mathfrak {A}$ is completely representable over $U$ iff $|U|= \kappa$, where $\kappa$ is the number of points of $\mathfrak {A}$. 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 ${x_0}; \ldots ;{x_{\alpha - 1}}$, in case one of the ${x_\kappa }{\text {’s}}$ is $1$. This result is proved in a general setting for a special class of groupoids.