Dictionary

identity element en · NOUN

Meanings

1. An element of an algebraic structure which when applied, in either order, to any other element via a binary operation yields the other element.
   • Therefore the number 0 is not considered an identity element for subtraction, even though x-0#61;x for all x, since 0-x#92;nex. 1990, Daniel M. Fendel, Diane Resek, Foundations of Higher Mathematics, Volume 1, Addison-Wesley, page 269:
   • Proof. If e and e' are both identity elements, then we have ee'=e since e' is an identity element, and ee'=e' since e is an identity element. Thus e=ee'=e'. 2003, Houshang H. Sohrab, Basic Real Analysis, Birkhäuser, page 17, Let (G,·) be a group. Then the identity element e∈G is unique. […]
   • Sometimes, to avoid ambiguity, we may use the notation e#95;M for the identity element of M. If multiplicative notation is used then we use the term identity element, and often use the notation 1, or 1#95;M, for the neutral element e. 2015, Martyn R. Dixon, Leonid A. Kurdachenko, Igor Ya. Subbotin, An Introduction to Essential Algebraic Structures, Wiley, page 41:

Forms

Hyponyms

zero (element that when applied with a binary operation leaves any other element unchanged) · additive identity (element that when applied with a binary operation leaves any other element unchanged) · multiplicative identity (element that when applied with a binary operation leaves any other element unchanged) · zero element (element that when applied with a binary operation leaves any other element unchanged)

Relateds

right identity · left identity · unity

Synonyms

neutral element (element that when applied with a binary operation leaves any other element unchanged) · identity (element that when applied with a binary operation leaves any other element unchanged)

Translations (16)

ga ball ionannais (member of a structure) · ru тожде́ственность (member of a structure) · fr élément neutre (member of a structure) · fr élément identité (member of a structure) · fi identiteettialkio (member of a structure) · it elemento neutro (member of a structure) · de Einheitselement (member of a structure) · mi tūmau (member of a structure) · nl identiteit (member of a structure) · hu egységelem (member of a structure) · pl element neutralny (member of a structure) · ru то́ждество (member of a structure) · sv identitetselement (member of a structure) · de neutrales Element (member of a structure) · he איבר יחידה (member of a structure) · es elemento neutro (member of a structure)