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Some varieties containing relation algebras


Author: Roger Maddux
Journal: Trans. Amer. Math. Soc. 272 (1982), 501-526
MSC: Primary 03G15; Secondary 03G25, 06E99, 08C10
DOI: https://doi.org/10.1090/S0002-9947-1982-0662049-7
MathSciNet review: 662049
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Abstract: Three varieties of algebras are introduced which extend the variety $RA$ of relation algebras. They are obtained from $RA$ by weakening the associative law for relative product, and are consequently called nonassociative, weakly-associative and semiassociative relation algebras, or $NA$, $WA$, and $SA$, respectively. Each of these varieties arises naturally in solving various problems concerning relation algebras. We show, for example, that $WA$ is the only one of these varieties which is closed under the formation of complex algebras of atom structures of algebras, and that $WA$ is the closure of the variety of representable $RA$’s under relativization. The paper also contains a study of the elementary theories of these varieties, various representation theorems, and numerous examples.


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Keywords: Relation algebra, Boolean algebra with operators, relativization, complex algebra
Article copyright: © Copyright 1982 American Mathematical Society