Linear growth for semigroups which are disjoint unions of finitely many copies of the free monogenic semigroup

TL;DR

This paper proves that semigroups formed by finitely many disjoint copies of the natural numbers under addition exhibit linear growth, leading to their semigroup algebras being PI algebras.

Contribution

It establishes that such semigroups always have linear growth and that their semigroup algebras are PI algebras, a new result in semigroup theory.

Findings

Semigroups as finite disjoint unions of copies of natural numbers have linear growth.

Their semigroup algebras satisfy polynomial identities (PI).

The result links semigroup structure to algebraic properties.

Abstract

We show that every semigroup which is a finite disjoint union of copies of the free monogenic semigroup (natural numbers under addition) has linear growth. This implies that the the corresponding semigroup algebra is a PI algebra.