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Inequivalent representations of geometric relation algebras

Published online by Cambridge University Press:  12 March 2014

Steven Givant
Steven Givant*
Affiliation:
Department of Mathematics and Computer Science, Mills College, 5000 Macarthur Boulevard, Oakland, CA 94613, USA, E-mail: [email protected]
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Abstract

It is shown that the automorphism group of a relation algebra constructed from a projective geometry P is isomorphic to the collineation group of P. Also, the base automorphism group of a representation of over an affine geometry D is isomorphic to the quotient of the collineation group of D by the dilatation subgroup. Consequently, the total number of inequivalent representations of , for finite geometries P, is the sum of the numbers

where D ranges over a list of the non-isomorphic affine geometries having P as their geometry at infinity. This formula is used to compute the number of inequivalent representations of relation algebras constructed over projective lines of order at most 10. For instance, the relation algebra constructed over the projective line of order 9 has 56,700 mutually inequivalent representations.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 68 , Issue 1 , March 2003 , pp. 267 - 310
Copyright
Copyright © Association for Symbolic Logic 2003

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References

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