identity element

Noun

identity element (plural identity elements)

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. […]

   Synonyms:

   identity, neutral element