Characterization of m-potent Elements in HypG(n) for Some Natural Numbers n

Abstract

A generalized hypersubstitution of type τ = (n) is a function σ which maps the n-ary operation symbol f to the term σ(f) which does not necessarily preserve the arity. Let HypG(τ ) be the set of all these generalized hypersubstitutions of type (τ ). The set HypG(τ ) with the binary operation “◦G” and the identity generalized hypersubstitution forms a monoid. In studying semigroup theory, the index and the period of an element a of a finite semigroup are the smallest values of m ≥ 1 and r ≥ 1 such that am+r = am. An element with the index m and the period 1 is called an m-potent element. In this thesis, we characterize m-potent elements in this monoid HypG(n) for some natural number n.