Leftzero

Leftzero, in mathematical usage, typically refers to the left-zero semigroup. This is a semigroup (S, ·) with the property that the product of any two elements equals the left factor: for all a, b in S, a · b = a. The definition makes the right operand irrelevant in the product.

As a consequence, the operation is associative: (a · b) · c = a · c = a, and a · (b · c) = a · b = a. Every element is idempotent since a · a = a. The structure is generally noncommutative unless the set S has only one element. Because there is no general identity element unless S is a singleton, a left-zero semigroup is not a monoid except in that trivial case.

Examples can be constructed on any nonempty set S by defining a · b = a for all a, b in S; the resulting algebraic structure is a left-zero semigroup. This contrasts with the right-zero semigroup, where the product is defined as a · b = b.

Relation to broader theory: a left-zero semigroup is a specific instance of a rectangular band and serves as a simple, idempotent, noncommutative example in semigroup theory. It helps illustrate how a binary operation can be determined solely by the left factor while still meeting associativity.

In literature, the term leftzero or left-zero semigroup is primarily used within algebra to denote this construction. Outside of pure algebra, the term may appear as a name or handle in various contexts, but the mathematical concept centers on the left-projection product.