Weaker Universalities in Semigroup Varieties

A variety V has an alg-universal n-expansion if the addition of n nullary operations to algebras from V produces an alg-universal category. It is proved that any semigroup variety V containing a semigroup that is neither an inflation of a completely simple semigroup nor an inflation of a semilattice of groups has an alg-universal 3-expansion. We say that a variety V is var-relatively alg-universal if for some proper subvariety W of V there is a faithful functor F from the category of all digraphs and compatible mappings into V such that Im(Ff) belongs to W for no compatible mapping f and if f : FG → FG′ is a homomorphism where G and G′ are digraphs then either Im(f) belongs to W or f = Fg for a compatible mapping g : G → G′. For a cardinal α ≥ 2, a variety V is α-determined if any set A of V-algebras of cardinality α such that the endomorphism monoids of A and B are isomorphic for all A, B ∈ A contains distinct isomorphic algebras. Similar sufficient conditions for a semigroup variety V to be α-determined for no cardinal α or var-relatively alg-universal are given. These results are proved by an analysis of three specific semigroup varieties. AMS Mathematics Subject Classification (2000): 20M07, 18B15, 08B15