Monoid atom

This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
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Definition

An atom in a monoid is an element in the monoid that cannot be expressed as a product of nonidentity elements of the monoid.

Related notions

Atomic monoids

A monoid is said to be an atomic monoid 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