All solid varieties of semigroups (Q1305436)

An identity \(u=v\) is said to be a hyperidentity in a variety \(\mathcal V\) if \(\mathcal V\) hypersatisfies \(u=v\) in the sense that whenever the operation symbols occurring in the terms \(u\) and \(v\) are replaced by any term (of the same type) of the appropriate arity, \(\mathcal V\) satisfies the resulting identity. A variety is called solid if it hypersatisfies all its identities. In the class of all semigroups, the greatest solid variety \(\mathcal H\) has been discovered by \textit{K. Denecke} and \textit{J. Koppitz} [Semigroup Forum 49, No. 1, 41-48 (1994; Zbl 0806.20049)]; \(\mathcal H\) is exactly the variety that hypersatisfies the associative law. The author [Algebra Universalis 36, No. 3, 363-378 (1996; Zbl 0905.20038)] has given a simple equational basis and an efficient solution to the word problem for \(\mathcal H\). In the present paper he characterizes all solid varieties of semigroups. Namely, a non-trivial variety \({\mathcal V}\subseteq{\mathcal H}\) is solid if and only if \(\mathcal V\) contains all rectangular bands, is right-left dual and either consists of bands or contains a non-trivial zero multiplication semigroup.