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Idempotent tree languages

Abstract A tree language of a fixed type τ is any set of terms of type τ. We consider here a binary operation +n on the set Wτ (Xn) of all n-ary terms of type τ, which results in semigroup (Wτ (Xn), +n). We characterize languages which are idempotent with respect to this binary operation, and look at varieties of tree languages containing idempotent languages. We also compare properties of semigroup homomorphisms from (𝒫(Wτ (Xn));+n) to (𝒫(Wτ (Xm));+m) with properties of homomorphisms between the corresponding absolutely free algebras ℱτ (Xn) and ℱτ (Xm).