Einselement

Einselement, in mathematics, refers to the identity element of a binary operation. It is an element that leaves other elements unchanged when combined with them under the given operation. Formally, let S be a set with a binary operation ◦: S × S → S. An element e ∈ S is called the Einselement if, for every a ∈ S, e ◦ a = a and a ◦ e = a. When an operation has only a left identity or only a right identity, that element is referred to as a left identity or right identity, respectively. In structures where a two-sided Einselement exists for all elements, it is unique.

Existence and uniqueness: If a two-sided Einselement exists, it is unique. If a left identity and a right identity both exist for every element, they coincide and yield the same Einselement.

Examples: In the integers under addition (Z, +), the Einselement is 0, since 0 + a = a and a + 0 = a for all a. In the real numbers under multiplication (R, ×), the Einselement is 1, since 1 · a = a and a · 1 = a for all a. For function composition on the set of all functions from X to X, the identity function id_X serves as the Einselement. In matrix algebra with standard multiplication, the identity matrix I serves as the Einselement.

Significance: The Einselement is central to the structure of algebraic systems such as monoids and groups. A monoid is a set with an associative binary operation and an Einselement; a group requires that every element also has an inverse with respect to that operation. In rings, the multiplicative Einselement is often denoted as 1 and the additive Einselement as 0, reflecting two distinct identity concepts within the same algebraic framework.