All \(2\)-solid varieties of semigroups
For a given type \(\tau\), a mapping from a set of operation symbols into the set of all terms of type \(\tau\) is called a hypersubstitution. The set \(\text{Hyp}(\tau)\) of all hypersubstitutions is a monoid with respect to composition and the identity map. For a variety \(V\) of type \(\tau\) an equivalence relation \(\sim_V\) on \(\text{Hyp}(V)\) of \(V\)-equivalent substitutions is a congruence if and only if \(V\) has a certain solidity property which for semigroup varieties is known to be equivalent to a condition known as 2-solidity. This paper characterizes all 2-solid varieties of semigroups as of two types: (i) band varieties containing all rectangular bands and (ii) varieties \(V\) containing the variety defined by the identity \(xyz=xz\), contained in another known finitely based variety \(W_2\), and whose set of identities is closed under duals of its 2-variable identities.
scientific article; zbMATH DE number 1445559