Valuations and hypersubstitutions (Q1430464)

A valuation of a term \(t\) is a non-negative integer which expresses in a certain sense the complexity of \(t\) (it can be the number of operations occurring in \(t\), the number of variables, etc). A hypersubstitution \(\sigma\) is called increasing with respect to a valuation \(v\) if \(v(t)\leq v(\overline\sigma(t))\). This determines a Galois connection between the set of all hypersubstitutions of type \(\mathcal T\) and the set \(\text{Val}({\mathcal T})\) of all valuations of type \(\mathcal T\). The result is applied to produce a monoid \(N\) for which the \(k\)-normalized variety of an \(M\)-solid variety \(\mathcal V\) with respect to a given valuation is \(N\)-solid.