Essential variables in hypersubstitutions.
Given a type \(\tau = (n_i)_{i\in I}\), a hypersubstitution is a mapping \(\sigma \) which assigns to every \(n_i\)-ary operation symbol \(f_i\) an \(n_i\)-ary term \(\sigma (f_i)\). Such a mapping can be inductively extended to the set of all terms of type \(\tau \). Let \(\mathcal V\) be a variety of type \(\tau \) and \(\sigma _t:f \to t\) be a hypersubstitution. The variable \(x_i\) is called essential in \(\sigma _t\) if \(x_i\) is essential in the term \(t\) with respect to \(\mathcal V\). Let \({\mathcal M_i}({\mathcal V})\) be the set of all hypersubstitutions \(\sigma _t\) such that \(x_i\) is essential in \(\sigma _t\). Consider a variety \(\mathcal V\) of type \((n)\). The authors give a complete answer whether \({\mathcal M_i}({\mathcal V})\) forms a monoid. This is important since to every monoid of hypersubstitutions there corresponds a complete sublattice of the lattice of subvarieties. For varieties of semigroups they characterize both \({\mathcal M_1}({\mathcal V})\) and \({\mathcal M_2}({\mathcal V})\).
scientific article; zbMATH DE number 2158753