1element

1element is a term encountered in mathematics and theoretical computer science contexts to denote either a singleton element or the identity element of an algebraic structure; the exact meaning depends on context. In set theory, a 1element set refers to a singleton, a set containing exactly one element. For any element x, the set {x} has cardinality one; singletons are foundational in constructions such as ordered pairs and in defining functions via images and preimages.

In algebra, the term 1element is sometimes used as a shorthand for the multiplicative identity, commonly denoted 1, which satisfies a·1 = a for all elements a in a ring, field, or monoid. The identity element acts as a neutral element for the operation and plays a central role in defining units and invertibility within the structure.

From a category-theory perspective, in the category of sets, a one-element set, often denoted 1, is a terminal object, meaning there is exactly one morphism from any set to 1. This abstracts the notion of a unique constant value and generalizes to other categories, where a terminal object serves a similar universal property.

Examples include the set {0} as a 1element set, and the number 1 in the integers acting as the multiplicative identity. The empty set is not a 1element set.

Notes: The term 1element is not widely standardized; readers typically encounter the explicit terms singleton and identity element. See also singleton, identity element, terminal object.