Nonelementary inclusive varieties of groups and semigroups

The class of identical inclusions was defined by E. S. Lyapin. This is the class of universal formulas which is situated strictly between identities and universal positive formulas. Classes of semigroups defined by identical inclusions are called inclusive varieties. Inclusive varieties that cannot be defined by the first order formulas are called nonelementary inclusive varieties. We study nonelementary inclusive varieties of groups, Clifford semigroups and nilsemigroups. In particular, a criterion for an inclusive variety to be nonelementary is found and limit nonelementary inclusive varieties of abelian groups are described. We also describe the upper semilattice of nonelementary inclusive varieties of finite abelian groups and prove that it is uncountable. We find an uncountable set of nonelementary inclusive varieties of nilpotent class 3 and nil class 2 finite commutative semigroups and a limit nonelementary inclusive variety of nilsemigroups. We consider completely regular semigroups in semigroup signature with an additional unary operation and nilsemigroups in semigroup signature with the additional constant 0.