# A Pseudo Representation Theorem for Various Categories of Relations

## M. Winter

It is well-known that, given a Dedekind category {\cal R} the category of (typed) matrices with coefficients from {\cal R} is a Dedekind category with arbitrary relational sums. In this paper we show that under slightly stronger assumptions the converse is also true. Every atomic Dedekind category {\cal R} with relational sums and subobjects is equivalent to a category of matrices over a suitable basis. This basis is the full proper subcategory induced by the integral objects of {\cal R}. Furthermore, we use our concept of a basis to extend a known result from the theory of heterogeneous relation algebras.

Keywords: Relation Algebra, Dedekind category, Allegory, Representability, Matrix Algebra.

2000 MSC: 18D10,18D15,03G15.

Theory and Applications of Categories, Vol. 7, 2000, No. 2, pp 23-37.

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