Identity Elements

Why

We can construct functions on the ground set of an algebra by fixing an element in the ground set and defining a function which maps elements to the result of the operation applied to the fixed element and the given element.

Definition

Let $(A, +)$ be an algebra. For each $a \in A$, denote by $+_a: A \to A$ the function defined by

\[ +_a(b) = a + b. \]

If $+_a$ is the identity function on $A$ then we call $a$ a left identity element of the algebra.

Similarly, denote by $+^a: A \to A$ the function defined by

\[ +^{a}(b) = b + a. \]

If $+^a$ is the identity function on $A$ then we call $a$ a right identity element of the algebra.

An identity element of the algebra is an element which is both a left and right identity. If the operation commutes, then a left identity and right identities are the same.