Substructures of a semigroup of tree languages

The set of all terms of type [Formula: see text] is denoted by [Formula: see text] Each element of [Formula: see text] is called a tree language. A binary operation [Formula: see text] is an operation on [Formula: see text] defined from the superposition [Formula: see text] on the set of all [Formula: see text]-ary terms of type [Formula: see text] Since [Formula: see text] satisfies the superposition law, [Formula: see text] is a semigroup. In this paper, we study subsemigroups of the semigroup [Formula: see text] including constant, left-zero, right-zero subsemigroups, rectangular bands, subgroups, and inverse subsemigroups. We also study factorization and locally factorization of the semigroup [Formula: see text] and we obtain that this semigroup is neither factorizable nor locally factorizable. However, we achieve a necessary and sufficient condition for factorization of local subsemigroup [Formula: see text], where [Formula: see text] is idempotent in [Formula: see text]. Finally, we study [Formula: see text]iteration and show that with special conditions, an [Formula: see text]iteration of idempotent is idempotent.