Almost universal varieties of monoids (Q801164)

A variety V of monoids is said to be almost universal if every category of algebras is isomorphic to the class of all nonzero homomorphisms between members of V. It is proved that a variety V of monoids is almost universal iff it is balanced and fails to satisfy the identity \(x^ ny^ n=(xy)^ n\) for all \(n>1\). Also, V is almost universal iff every group with zero is isomorphic to End(A) for some \(A\in V\).