# Atomic monoid

From Groupprops

*This article defines a monoid property, viz a property that can be evaluated for any monoid. Recall that a monoid is a set with an associative binary operation, having a neutral element (viz multiplicative identity)*

## Definition

### Definition with symbols

An **atom** in a monoid is an element in the monoid that cannot be expressed as a product of nonidentity elements of the monoid. A monoid is said to be atomic if:

- Every element can be expressed as a product of atoms
- For every element, the supremum of lengths of all possible words in the atoms that can be used to express it, is finite