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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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