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.

A superposition is an operation of terms by which we substitute each variable within a term with other forms of terms. With more options of terms to be replaced, an inductive superposition is apparently more general than the superposition. This comes with a downside that it does not satisfy the superassociative property on the set of all terms of a given type while the superposition does. A derived base set of terms on which the inductive superposition is superassociative is given in this paper. A clone-like algebraic structure involving such base set and superposition is the main topic of this paper. Generating systems of the clone-like algebra are characterized and it turns out that the algebra is only free with respect to itself under the certain selections of fixed terms concerning its inductive superposition or the specific type of its base set.

In universal algebra, it is well-known that if $S$ is an algebraic structure, then the kind of algebraic structure of $S/\rho$ is similar to $S$ where $\rho$ is a congruence relation on $S$. In this work, we study the notion of a full $k$-ideal $A$ of an $n$-ary semiring $S$ and construct a congruence relation $\rho$ on $S$ with respect to the full $k$-ideal $A$ in order to make the quotient $n$-ary semiring $S/\rho$ to be an $n$-ary ring. Moreover, the notion of an $h$-ideal of an $n$-ary semiring was studied and connections between an $h$-ideal and a $k$-ideal of an $n$-ary semiring were investigated.