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Pair-dense relation algebras


Author: Roger D. Maddux
Journal: Trans. Amer. Math. Soc. 328 (1991), 83-131
MSC: Primary 03G15; Secondary 08B99, 68Q99
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 $ \{ \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 $ \vert U\vert= \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 $ \vert U\vert= \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.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1991-1049616-1
Keywords: Relation algebra, semiassociative relation algebra, representable, completely representable, associativity, groupoid
Article copyright: © Copyright 1991 American Mathematical Society