Abstract
The constant mappings onto the unit form a zero subcategory of any category of monoid homomorphisms; a varietyV of monoids isalmost universal if every category of algebras is isomorphic to a class of all nonzero homomorphisms between members ofV. Almost universal monoid varieties are shown to be exactly those varieties containing all commutative monoids in which the identity xnyn=(xy)n fails for every n>1. Almost universal varieties of monoids can also be characterized categorically as the varieties containing all groups with zero as one-object full subcategories.
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