Endomorphism monoids of bands

Abstract

Endomorphisms of any algebra A form a monoid End(A) under their composition. In this paper we investigate the endomorphism monoids of bands, and the question of when End(A) determines A6V within a variety V of bands. An early result on determination by endomorphisms not of algebras but partially ordered sets is due to L.M. Gluskin [7]. It says that two partially ordered sets S and T whose endomorphism semigroups are isomorphic must themselves be either isomorphic or antiisomorphic. Analogous results hold for some small varieties of algebras, for example, for a variety of distributive lattices, see B.M. Schein []9], R. McKenzie and C. Tsinakis [14], and by P. Ribenboim [18]; for distributive p-algebras see [i], for Boolean rings K.D. Magill [13], for semilattices [19], and for Brouwerian semilattices [i0] and [21]. In all of these varieties there exist at most two non-isomorphic algebras with isomorphic endomorphism monoids. B.M. Schein [20] proved that there exist at most four non-isomorphic normal bands with ismorphic endomorphism monoids. Our aim is to generalize the results of B.M. Schein [19,20] to larger