Inequivalent representations of geometric relation algebras
by
Steven Givant
Mills College

In a 1961 article, Roger Lyndon - building on earlier work of Bjarni Jonsson - showed how to construct a relation algebra B(P) from any projective geometry P. When P is Desarguesian (for instance, when P has dimension at least 3), Lyndon showed that B(P) can be represented as an algebra of binary relations; in fact, he showed that there is a unique complete representation, up to equivalence - i.e., up to renaming the elements of the base set. (A representation is ``complete" if it preserves all infinite joins as unions. For finite geometries P, every representation of B(P) is necessarily complete.) When P is a non-Desarguesian plane, an argument due to Jonsson shows that P cannot be represented as - i.e., is not isomorphic to - a (set-theoretically defined) algebra of binary relations. For the case of a projective line P of order n, Lyndon showed that B(P) has a complete representation if and only if there is a projective plane of order n. However, he did not investigate the number of inequivalent complete representations that B(P) can have in this case.

In this talk, we present a general formula for computing the exact number of inequivalent complete representations of the algebras B(P). We use it to compute the number of inequivalent representations of the algebras for projective lines of orders n < 11 (the only orders for which the requisite information about projective planes of order n is currently known). For example, there are 120 inequivalent representations of B(P) when P has order 7, there are 240 when P has order 8, and there are 56, 700 when P has order 9. The formula can also be used to give lower bounds on the number of inequivalent representations when the requisite information about projective planes of order n is not known. For instance, there are at least 1.0888869 times 10^28 inequivalent representations of B(P) when P has order 29.

Date received: January 6, 2002

Copyright © 2002 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caig-55.