Cardinalvalued

Cardinalvalued is an adjective used in mathematics to describe objects, functions, or invariants whose values are cardinal numbers rather than real numbers or ordinals. In set theory, a function is cardinal-valued if its codomain consists of cardinals—the initial ordinals that index the sizes of sets. Informally, a cardinal-valued function assigns to each input a size.

Examples include the cardinality function that maps any set to its cardinality, denoted |A|. More generally, many size-measuring constructions in combinatorics, model theory, and algebra yield cardinal-valued invariants: for instance, the number of elements in a substructure, the size of a family of sets, or the number of elements in a quotient by a given equivalence relation.

Cardinal arithmetic governs the behavior of cardinal-valued quantities. For infinite cardinals κ and λ, one has κ + λ = max{κ, λ} and κ · λ = max{κ, λ}. These properties imply that many operations on cardinal-valued data preserve the cardinality type and that large-scale size comparisons can be simplified using the maximum operation.

The term is not always standard in published literature; it is often used informally to denote a quantity that takes cardinal values. It should be distinguished from ordinal-valued or real-valued functions, which output ordinals or real numbers respectively.

See also: cardinality, cardinal arithmetic, cardinal function, cofinality.