Algebra universalis

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Subcompletions of representable relation algebras

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  1. In memory of Bjarni Jónsson

Abstract

The variety of representable relation algebras is closed under canonical extensions but not closed under completions. What variety of relation algebras is generated by completions of representable relation algebras? Does it contain all relation algebras? It contains all representable finite relation algebras, and this paper shows that it contains many non-representable finite relation algebras as well. For example, every Monk algebra with six or more special elements (called “colors”) is a subalgebra of the completion of an atomic symmetric integral representable relation algebra whose finitely-generated subalgebras are finite.

Keywords

Relation algebra Representable relation algebra Monk algebra Completion Canonical extension Cylindric algebra 

Mathematics Subject Classification

03G15 

Notes

Acknowledgements

The most direct inspiration for this work is [15]; see also [12, 13, 16, 19], and especially [14]. The method used in a proof of representability in [15] was embodied in the notion of flexible trio. A modification of a construction from [15] causes the finitely-generated subalgebras to be finite. Suggestions for the form and content of the Introduction came from the referee and the editors, to whom I express my thanks.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsIowa State UniversityAmesUSA

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