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On Endoprimal Monoids in Clone Theory

The endomorphisms of a universal algebra on a universe A form a monoid of selfmaps on A. Such monoids, called endoprimal monoids, are special and rare. In this paper we study the first interesting case of a 3-element set A. In the lattice of inclusion-ordered endoprimal monoids we find three maximal endoprimal monoids. This knowledge allows us to conclude that many large monoids are not endoprimal.