Abstract
The purpose of this paper is to start a general investigation of the varieties "P'(M) obtained by expanding a variety T" by a monoid of endomorphisms M. This construction was used in [3] to manufacture the first example of a variety with a decidable theory and not of the form (discriminator)| It also plays a key role in Baur's papers [1], [2] on the first-order theory of Abelian groups with distinguished subgroups. In the first section a few basic results are presented. In the second section we describe exactly when 7/'(M) is a discriminator variety, generalizing the treatment of Nsg(G) given in [3]. The final section is devoted to Abelian varieties and the corresponding varieties of modules.
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