英和和英

名詞

identity element (複数形 identity elements)

1. (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 ${\displaystyle 0}$ is not considered an identity element for subtraction, even though ${\displaystyle x-0=x}$ for all ${\displaystyle x}$, since ${\displaystyle 0-x\neq x}$.

   • 2003, Houshang H. Sohrab, Basic Real Analysis, Birkhäuser, page 17,

     Let ${\displaystyle (G,\cdot )}$ be a group. Then the identity element ${\displaystyle e\in G}$ is unique. […]

     Proof. If ${\displaystyle e}$ and ${\displaystyle e'}$ are both identity elements, then we have ${\displaystyle ee'=e}$ since ${\displaystyle e'}$ is an identity element, and ${\displaystyle ee'=e'}$ since ${\displaystyle e}$ is an identity element. Thus

     ${\displaystyle e=ee'=e'}$.

   • 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 ${\displaystyle e_{M}}$ for the identity element of ${\displaystyle M}$.

     If multiplicative notation is used then we use the term identity element, and often use the notation ${\displaystyle 1}$, or ${\displaystyle 1_{M}}$, for the neutral element ${\displaystyle e}$.