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 W τ ( X n ) of all n -ary terms of type τ , which results in semigroup ( W τ ( X n ), + 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 (?( W τ ( X n ));+ n ) to (?( W τ ( X m ));+ m ) with properties of homomorphisms between the corresponding absolutely free algebras ℱ τ ( X n ) and ℱ τ ( X m ).
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