Hagemann’s theorem for regular categories

Abstract

In this paper we extend the characterisation of n-permutable varieties of universal algebras due to J. Hagemann to regular categories. In particular, we show that a regular category has n-permutable congruences if and only if every internal reflexive relation R in it satisfies R^\circ \leqslant R^{n-1}, and if and only if every internal reflexive relation R in it satisfies R^n\leqslant R^{n-1}. In the case when n=2 this result is well known.