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.