Inclusive varieties of Clifford semigroups

Abstract The class of identical inclusions was defined by E. S. Lyapin. This class of nonelementary universal formulas is situated strictly between identities and universal positive formulas. These formulas can be written as identical equalities of subsets of a free algebra, in particular, of $$X^+$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>X</mml:mi> <mml:mo>+</mml:mo> </mml:msup> </mml:math> or of a free unary semigroup. Classes of semigroups defined by identical inclusions are called inclusive varieties . We describe the lattice of inclusive varieties of Clifford semigroups modulo the lattice of inclusive varieties of groups and show that even in the lowest layers of the lattice of inclusive varieties of Abelian groups there exist continual parts. We characterize Clifford semigroups defined by their identical inclusions up to isomorphism. A partial classification of nonperiodic and of reducible Abelian groups up to inclusive equivalence and a list of groups from this class definable by its identical inclusions up to isomorphism are obtained. We consider a syntactic presentation of all inclusive varieties of Clifford semigroups over groups from an inclusive variety of groups.