Group of units of a monoid: Difference between revisions
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===In symbols=== | ===In symbols=== | ||
Let <math>M</math> be a monoid. Then <math>G=\{ m \in M: \exists n \in M: nm=e \}</math>. Then <math>G</math> is a group, the group of units of a monoid. | Let <math>M</math> be a [[monoid]]. Then <math>G=\{ m \in M: \exists n \in M: nm=e \}</math>. Then <math>G</math> is a [[group]], the '''group of units''' of a monoid. | ||
==Examples== | |||
One simple family of examples: The [[group of units modulo n]] is the group of units for the [[monoid of integers modulo n under multiplication]]. | |||
Latest revision as of 01:32, 12 January 2024
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
In words
Given a monoid, its group of units is the set of all invertible elements in the monoid under the monoid's operation. It is indeed a group.
In symbols
Let be a monoid. Then . Then is a group, the group of units of a monoid.
Examples
One simple family of examples: The group of units modulo n is the group of units for the monoid of integers modulo n under multiplication.