An Inductive Product of Terms

Abstract

Over the years, many binary operations on the set of all $n$-ary terms of type $\tau$ have been defined, derived from superpositions and forming semigroups. Later, an inductive composition of terms was introduced as a generalization of a superposition, extending its scope from variable replacement to subterm replacement. From this, a binary operation called an $r$-inductive product was introduced by fixing a specific subterm to be replaced. In this study, we define a new binary operation, called an $rs$-inductive product, which generalizes the $r$-inductive product by allowing the simultaneous replacement of two specific subterms. We construct a semigroup equipped with the new operation and investigate its algebraic properties, including regular elements, idempotent elements, and Green’s relations.