   ISSN 1088-6850(online) ISSN 0002-9947(print)

Pair-dense relation algebras

Journal: Trans. Amer. Math. Soc. 328 (1991), 83-131
MSC: Primary 03G15; Secondary 08B99, 68Q99
DOI: https://doi.org/10.1090/S0002-9947-1991-1049616-1
MathSciNet review: 1049616
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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 (with allowed). In a simple pair-dense relation algebra, every pair is either a "point" (an algebraic analogue of ) or a "twin" (a pair which contains no point). In fact, every simple pair-dense relation algebra is completely representable over a set iff , where is the number of points of and is the number of twins of .

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 is completely representable over iff , where is the number of points of . 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 may be characterized as a simple complete point-dense semiassociative relation algebra whose set of points has the same cardinality as .

Semiassociative relation algebras may not be associative, so the equation may fail, but it does hold if any one of , or is . In fact, any rearrangement of parentheses is possible in a term of the form , in case one of the is . This result is proved in a general setting for a special class of groupoids.

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