Math 2.0

Group theory and generalizations Definition

Group of units of a monoid

Builds on MonoidInverse element

Definition

Let (M,∗,e)(M, \ast, e) be a monoid. An element u∈Mu \in M is a unit when it has a two-sided inverse, that is, an element v∈Mv \in M with u∗v=v∗u=eu \ast v = v \ast u = e. The group of units of MM is the set M×M^\times of the units of MM, with the operation ∗\ast restricted to M×M^\times. It is a group with identity ee.

Remarks

The units of the monoid (ℤ,⋅)(\mathbb{Z}, \cdot) are 11 and −1-1. The units of the monoid of n×nn \times n real matrices under multiplication are the invertible matrices, and they form the general linear group GLn(ℝ)\mathrm{GL}_n(\mathbb{R}). For a ring RR, the group of units R×R^\times is the group of units of the monoid (R,⋅)(R, \cdot).

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Binary operation, definition, no evidence levelBinary operationdefinitionAssociativity, definition, no evidence levelAssociativitydefinitionIdentity element, definition, no evidence levelIdentity elementdefinitionSemigroup, definition, no evidence levelSemigroupdefinitionInverse element, definition, no evidence levelInverse elementdefinitionMonoid, definition, no evidence levelMonoiddefinitionGroup of units of a monoid, definition, no evidence levelGroup of units of a m…definition
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  1. Binary operation definition
  2. Associativity definition
  3. Identity element definition
  4. Semigroup definition
  5. Inverse element definition
  6. Monoid definition
  7. Group of units of a monoid definition

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Further reading

  1. Web https://en.wikipedia.org/wiki/Unit_(ring_theory), Group of units
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