On endomorphism monoids in varieties of bands

Abstract

It will be proved that every non-trivial variety \({\mathbb{V}}\) of bands (idempotent semigroups) contains a proper generating class of non-isomorphic bands B such that B generates \({\mathbb{V}}\) and any band \({B\prime}\) having the same endomorphism monoid as B is isomorphic to B or to the opposite band Bop. Consequently, every sharply greater band variety has a sharply greater class of endomorphism monoids.