On Universality of Semigroup Varieties

To Professor Jǐŕı Rosický on his 60th birthday Abstra t. A category K is called α-determined if every set of non-isomorphic Kobjects such that their endomorphism monoids are isomorphic has a cardinality less than α. A quasivariety Q is called Q-universal if the lattice of all subquasivarieties of any quasivariety of finite type is a homomorphic image of a sublattice of the lattice of all subquasivarieties of Q. We say that a variety V is var-relatively alguniversal if there exists a proper subvariety W of V such that homomorphisms of V whose image does not belong to W contains a full subcategory isomorphic to the category of all graphs. A semigroup variety V is nearly J -trivial if for every semigroup S ∈ V any J -class containing a group is a singleton. We prove that for a nearly J -trivial variety V the following are equivalent: V is Q-universal; V is var-relatively alg-universal; V is α-determined for no cardinal α; V contains at least one of the three specific semigroups. Dually, for a nearly J -trivial variety V the following are equivalent: V is 3-determined; V is not var-relatively alg-universal; the lattice of all subquasivarieties of V is finite; V is a subvariety of one of two special finitely generated varieties.