On substructures of semigroups of inductive terms

An inductive composition is an operation generalizing from a superposition $S^n$ on the set of all $n$-ary terms of type $\tau$. A binary operation called inductive product is obtainable from such composition. It is a generalization of a tree language product but on the set of all $n$-ary terms of type $\tau$. Unlike the original one, this inductive product is not associative on the mentioned set. Nonetheless, it turns to be associative on some restricted set. A semigroup arising in this way is the main focus of this paper. We consider its special subsemigroups and semigroup factorizations.