Atomic monoid

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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