# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Trans. Amer. Math. Soc. 328 (1991), 83-131 Request permission

## 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.
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