Multi-solid Varieties

We consider the concepts of colored terms and multihypersubstitutions. If t ∈ Wτ (X) is a term of type τ , then any mapping αt : Pos (t) → IN of the non-variable positions of a term into the set of natural numbers is called a coloration of t. The set W c τ (X) of colored terms consists of all pairs 〈t, αt〉. The set Hyp(τ) of all hypersubstitutions of type τ is a countable monoid. If M is a submonoid of Hyp(τ) then any sequence ρ = (σ1, σ2, . . .) is a mapping ρ : IN → M , called a multi-hypersubstitution over M . An identity t ≈ s, satisfied in a variety V is an M -multi-hyperidentity if its images ρ[t ≈ s] are also satisfied in V for all ρ ∈ 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. Pre-complete congruences are satisfied in the pre-trivial varieties. Studying the multi-hypersubstitutions we find out necessary and sufficient conditions a variety to be pre-complete. Finally we give an automata realization of multi-hypersubstitutions and colored terms. AMS 2000 Mathematics Subject Classification: 08B15, 03C05;