identity element

plural identity elements

1. algebraAn 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

   2 more examples

   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

   Synonyms: identity, neutral element