Journal of Symbolic Logic

Undecidability of representability as binary relations

Robin Hirsch and Marcel Jackson

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In this article we establish the undecidability of representability and of finite representability as algebras of binary relations in a wide range of signatures. In particular, representability and finite representability are undecidable for Boolean monoids and lattice ordered monoids, while representability is undecidable for Jónsson's relation algebra. We also establish a number of undecidability results for representability as algebras of injective functions.

Article information

J. Symbolic Logic, Volume 77, Issue 4 (2012), 1211-1244.

First available in Project Euclid: 15 October 2012

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03G15: Cylindric and polyadic algebras; relation algebras 20M30: Representation of semigroups; actions of semigroups on sets
Secondary: 20F10: Word problems, other decision problems, connections with logic and automata [See also 03B25, 03D05, 03D40, 06B25, 08A50, 20M05, 68Q70] 08A99: None of the above, but in this section

Relation algebra binary relations representability Boolean monoid lattice ordered monoid Brandt semigroup


Hirsch, Robin; Jackson, Marcel. Undecidability of representability as binary relations. J. Symbolic Logic 77 (2012), no. 4, 1211--1244. doi:10.2178/jsl.7704090.

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