Small and countable inclusive varieties of 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.These universal formulas can be written as identical equalities of subsets of \(X^+\) $X^{+}$ . Classes of semigroups defined by identical inclusions are called inclusive varieties. We describe finite inclusive varieties of semigroups and study countable inclusive varieties of semigroups.We also describe small inclusive varieties, that is, inclusive varieties with finite lattices of their inclusive subvarieties, of completely regular semigroups and study inclusive varieties of completely regular semigroups with countable lattices of their inclusive subvarieties