Weakly diagonal algebras and definable principal congruences

An algebra is called weakly diagonal if every subuniverse of its square contains the graph of an automorphism. We show that every variety generated by a finite algebra with no proper subalgebras has a weakly diagonal generator. The result is applied in several ways and, in particular, to show that every arithmetical affine complete variety of finite type has equationally definable principal congruences. This paper is dedicated to Walter Taylor. Received February 22, 2005; accepted in final form June 3, 2005. Work of the first author was supported by grant No. 5368 from The Estonian Science Foundation.