Math 2.0

Group theory and generalizations Definition

Semigroup

Builds on Binary operationAssociativity

Definition

A semigroup is a pair (S,∗)(S, \ast), where SS is a set and ∗\ast is a binary operation on SS that is associative. Thus (a∗b)∗c=a∗(b∗c)(a \ast b) \ast c = a \ast (b \ast c) for all a,b,c∈Sa, b, c \in S.

Remarks

The positive integers with addition form a semigroup that has no identity element. Each monoid and each group is a semigroup. Some authors allow the empty set as a semigroup, and others do not.

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Binary operation, definition, no evidence levelBinary operationdefinitionAssociativity, definition, no evidence levelAssociativitydefinitionSemigroup, definition, no evidence levelSemigroupdefinition
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Further reading

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