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THE VARIETY OF COSET RELATION ALGEBRAS

Published online by Cambridge University Press:  21 December 2018

STEVEN GIVANT  and
HAJNAL ANDRÉKA
STEVEN GIVANT
Affiliation:
DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE MILLS COLLEGE 5000 MACARTHUR BOULEVARD, OAKLAND, CA94613, USAE-mail:givant@mills.edu
HAJNAL ANDRÉKA
Affiliation:
ALFRÉD RÉNYI INSTITUTE OF MATHEMATICS HUNGARIAN ACADEMY OF SCIENCES REÁLTANODA UTCA 13-15, BUDAPEST1053, HUNGARYE-mail:andreka.hajnal@renyi.mta.hu
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Abstract

Givant [6] generalized the notion of an atomic pair-dense relation algebra from Maddux [13] by defining the notion of a measurable relation algebra, that is to say, a relation algebra in which the identity element is a sum of atoms that can be measured in the sense that the “size” of each such atom can be defined in an intuitive and reasonable way (within the framework of the first-order theory of relation algebras). In Andréka--Givant [2], a large class of examples of such algebras is constructed from systems of groups, coordinated systems of isomorphisms between quotients of the groups, and systems of cosets that are used to “shift” the operation of relative multiplication. In Givant--Andréka [8], it is shown that the class of these full coset relation algebras is adequate to the task of describing all measurable relation algebras in the sense that every atomic and complete measurable relation algebra is isomorphic to a full coset relation algebra.

Call an algebra $\mathfrak{A}$ a coset relation algebra if $\mathfrak{A}$ is embeddable into some full coset relation algebra. In the present article, it is shown that the class of coset relation algebras is equationally axiomatizable (that is to say, it is a variety), but that no finite set of sentences suffices to axiomatize the class (that is to say, the class is not finitely axiomatizable).

Keywords

03G1503C0503C6003E2020A1503C5208C10relation algebragroupnonrepresentable relation algebracoset relation algebravarietynonfinite axiomatizability
Type
Articles
Information
The Journal of Symbolic Logic , Volume 83 , Issue 4 , December 2018 , pp. 1595 - 1609
Copyright
Copyright © The Association for Symbolic Logic 2018 

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References

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