SHORT-SOLID SUPERASSOCIATIVE TYPE (n) VARIETIES

Abstract

Let M be a monoid of hypersubstitutions and V be a variety, both of a fixed type τ. V is said to be M-solid if the application of any hypersubstitution in M to any identity of V leads to an identity of V . For any monoid M, the collection of all M-solid varieties of type τ forms a sublattice of the lattice of all varieties of type τ. Thus studying M-solid varieties gives an approach to studying the lattice of all varieties of a given type. This approach has been used most successfully in the special case of type (2) and in particular for varieties of semigroups, where the lattice of all M-solid varieties has been fully characterized for M = Hyp(2) and various other choices of M. In this paper we extend this approach to type (n) varieties forn ≥ 3. Using the n-ary superassociative law as our analogue of associativity for the semigroup case, we investigate M-solid superassociative varieties of type (n) for two monoids M consisting of particular hypersubstitutions which we call short hypersubstitutions. For each of these monoids we find the smallest and largest M-solid superassociative varieties, and give complete characterizations of all such varieties. To obtain these characterizations we introduce a reduction in M-solidity testing based on the Green’s relations on monoids of hypersubstitutions, which we describe for our monoids.