Multi-solid varieties and Mh-transducers

Abstract

We consider the concepts of colored terms and multi-hypersubstitutions. If \(t\in W_\tau(X)\) is a term of type \(\tau\), then any mapping  \(\alpha_t:Pos^\mathcal{F}(t)\to  \mathbb{N}\) of the non-variable positions of a term into the set of natural numbers is called a coloration  of  \(t.\) The set \(W_\tau^c(X)\) of  colored terms consists of all pairs \(\langle t,\alpha_t\rangle.\) Hypersubstitutions are maps which assign to each operation symbol a term with the same arity. If \(M\) is a monoid of hypersubstitutions then any sequence \(\rho = (\sigma_1,\sigma_2,\ldots)\) is   a mapping \(\rho:\mathbb{N}\to M\), called a multi-hypersubstitution over \(M\).  An identity \(t\approx s\), satisfied in a variety \(V\) is an \(M\)-multi-hyperidentity if its images \(\rho[t\approx s]\) are  also satisfied in \(V\) for all \(\rho\in M\). A variety \(V\) is \(M\)-multi-solid, if all its identities are \(M-\)multi-hyperidentities. We prove a series of inclusions and equations concerning \(M\)-multi-solid varieties.  Finally we give an automata realization of multi-hypersubstitutions and colored terms.