identity element
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English
[edit]Noun
[edit]identity element (plural identity elements)
- (algebra) An element of an algebraic structure which when applied, in either order, to any other element via a binary operation yields the other element.
- 1990, Daniel M. Fendel, Diane Resek, Foundations of Higher Mathematics, Volume 1, Addison-Wesley, page 269:
- Therefore the number is not considered an identity element for subtraction, even though for all , since .
- 2003, Houshang H. Sohrab, Basic Real Analysis, Birkhäuser, page 17,
- Let be a group. Then the identity element is unique. […]
- Proof. If and are both identity elements, then we have since is an identity element, and since is an identity element. Thus
- .
- 2015, Martyn R. Dixon, Leonid A. Kurdachenko, Igor Ya. Subbotin, An Introduction to Essential Algebraic Structures, Wiley, page 41:
- Sometimes, to avoid ambiguity, we may use the notation for the identity element of .
If multiplicative notation is used then we use the term identity element, and often use the notation , or , for the neutral element .
Usage notes
[edit]For binary operation defined on a given algebraic structure, an element is:
- a left identity if for any in the structure,
- a right identity, for any in the structure,
- simply an identity element or (for emphasis) a two-sided identity if both are true.
Where a given structure is equipped with an operation called addition, the notation may be used for the additive identity. Similarly, the notation denotes a multiplicative identity.
Synonyms
[edit]- (element that when applied with a binary operation leaves any other element unchanged): identity, neutral element
Hyponyms
[edit]- (element that when applied with a binary operation leaves any other element unchanged): additive identity, multiplicative identity, zero, zero element
Related terms
[edit]Translations
[edit]member of a structure
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See also
[edit]Further reading
[edit]Identity matrix on Wikipedia.Wikipedia
Inverse element on Wikipedia.Wikipedia