On power varieties of semigroups
A semigroup identity \(u=v\) is said to be stable if each u,v is a product of distinct variables. Let V be a variety of semigroups. Then Stab V denotes the smallest variety containing V which is defined by stable identities. The power variety of semigroups is the variety PV generated by the semigroups of all subsets of each semigroup from some variety V. The problem of determination of the power variety PV associated with V is solved in the following way. First, find the variety Stab V. To obtain PV, take the join of Stab V with the variety of all semilattices. Stab V is actually the variety \(P'V\) generated by the semigroups of the nonempty subsets of each member of V. It follows that the operator \(P'\) is idempotent while \(P\neq P^ 2=P^ 3\). Further, \(P^ 2V\) is defined by the permutation identities which hold in V. Via the manipulation of stable identities and using some basic results from the theory of well- quasi-ordering, it is shown that the semilattice of varieties of the form PV is countable and better quasi-ordered under inclusion. Similar questions are also considered in the context of generalized varieties.
scientific article; zbMATH DE number 4112898