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Logic Journal of IGPL 2005 13(2):159-171; doi:10.1093/jigpal/jzi012
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© The Author, 2005. Published by Oxford University Press. All rights reserved.

Original Articles

The Class of Representable Ordered Monoids has a Recursively Enumerable, Universal Axiomatisation but it is Not Finitely Axiomatisable

Robin Hirsch

Department of Computer Science, University College London, Gower Street, London, UK. Email: R.Hirsch{at}cs.ucl.ac.uk

An ordered monoid is a structure with an identity element (1'), a binary composition operator (;) and an antisymmetric partial order (≤), satisfying certain axioms. A representation of an ordered monoid is a 1-1 map which maps elements of an ordered monoid to binary relations in such a way that 1' is mapped to the identity relation, ; corresponds to composition of binary relations and ≤ corresponds to inclusion of binary relations.

We devize a two player game that tests the representability of an ordered monoid n times and show that these games characterise representability. From this we obtain a recursively enumerable, universal axiomatisation of the class of all representable ordered monoids.

For each n < {omega} we construct an unrepresentable ordered monoid An and show that the second player has a winning strategy in a game of length n. Hence we prove that the class of all representable ordered monoids is not finitely axiomatisable.

Key Words: Binary relation, ordered monoid, partial order, game, non finitely axiomatisable

Received 28 September 2004. Revised 4 November 2004.


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