An Inductive Product of Terms (European Journal of Pure and Applied Mathematics, 2025)
Over the years, many binary operations on the set of all $n$-ary terms of type $τ$ 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.
Keywords: inductive product; inductive composition; semigroups; regular elements; idempotent elements; Green's relations; terms