Math 2.0

Group theory and generalizations Definition

Monoid

Builds on SemigroupIdentity element

Definition

A monoid is a triple (M,∗,e)(M, \ast, e), where (M,∗)(M, \ast) is a semigroup and e∈Me \in M is an identity element for ∗\ast. Thus (a∗b)∗c=a∗(b∗c)(a \ast b) \ast c = a \ast (b \ast c) and e∗a=a∗e=ae \ast a = a \ast e = a for all a,b,c∈Ma, b, c \in M.

Remarks

The natural numbers ℕ={0,1,2,…}\mathbb{N} = \{0, 1, 2, \ldots\} with addition form a monoid with identity 00. The integers with multiplication form a monoid with identity 11. The identity element of a monoid is unique, thus some authors write a monoid as the pair (M,∗)(M, \ast). Each group is a monoid in which each element has an inverse.

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Binary operation, definition, no evidence levelBinary operationdefinitionAssociativity, definition, no evidence levelAssociativitydefinitionIdentity element, definition, no evidence levelIdentity elementdefinitionSemigroup, definition, no evidence levelSemigroupdefinitionMonoid, definition, no evidence levelMonoiddefinition
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  1. Binary operation definition
  2. Associativity definition
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  4. Semigroup definition
  5. Monoid definition

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Further reading

  1. Web https://en.wikipedia.org/wiki/Monoid, Definition
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