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Pair-dense relation algebras. (English) Zbl 0746.03055
Summary: 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 \(x_ 0;\cdots;x_{\alpha-1}\), in case one of the \(x_ \kappa\)’s is 1. This result is proved in a general setting for a special class of groupoids.

MSC:
03G15 Cylindric and polyadic algebras; relation algebras
20L05 Groupoids (i.e. small categories in which all morphisms are isomorphisms)
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