Atomic monoid: Difference between revisions
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==Definition== | ==Definition== | ||
An [[Monoid atom|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: | An [[Monoid atom|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: | ||
Latest revision as of 23:31, 11 January 2024
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
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