nLab
GCD monoid

Contents

Context

Algebra

Monoid theory

Contents

Definition

A monoid (or semigroup) MM is a left GCD monoid if for all elements m,nMm,n \in M, there is a unique minimal left ideals in MM generated by mm and nn. Similarily, right GCD monoid if for all elements m,nMm,n \in M, there is a unique minimal right ideals in MM generated by mm and nn. Finally, MM is a two-sided GCD monoid if it is both a left GCD monoid and a right GCD monoid, and MM is a GCD monoid if MM is a two-sided GCD monoid and for all elements m,nMm,n \in M the minimal left and right ideals generated by mm and nn coincide.

Last revised on May 21, 2021 at 18:25:40. See the history of this page for a list of all contributions to it.