PlanetMath: identity element

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identity element

(Definition)

Let be a groupoid, that is a set with a binary operation $G \times G \to G$ , written muliplicatively so that $(x, y) \mapsto xy$ .

An identity element for is an element such that for all $g \in G$ .

The symbol is most commonly used for identity elements. Another common symbol for an identity element is , particularly in semigroup theory (and ring theory, considering the multiplicative structure as a semigroup).

Groups, monoids, and loops are classes of groupoids that, by definition, always have an identity element.

"identity element" is owned by mclase. [ full author list (3) | owner history (2) ]

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Other names:

neutral element

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Cross-references: classes, loops, monoids, groups, structure, ring, theory, semigroup, binary operation, groupoid
There are 106 references to this entry.

This is version 6 of identity element, born on 2002-06-27, modified 2002-10-10.
Object id is 3140, canonical name is IdentityElement.
Accessed 14728 times total.

Classification:

AMS MSC:	20A05 (Group theory and generalizations :: Foundations :: Axiomatics and elementary properties)
	20N02 (Group theory and generalizations :: Other generalizations of groups :: Sets with a single binary operation )
	20N05 (Group theory and generalizations :: Other generalizations of groups :: Loops, quasigroups)
	20M99 (Group theory and generalizations :: Semigroups :: Miscellaneous)

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