finites

Finites is a term that some writers use as a noun to refer to finite objects in mathematics and logic. In standard terminology, the adjective finite is used, as in finite sets, finite groups, or finite automata. When "finites" appears as a noun, it typically denotes a collection containing only finitely many elements or, more broadly, a class defined by finiteness. The usage is not common in formal texts, where the plural form of the noun is rarely employed.

In set theory and combinatorics, a finite set is a set with a finite cardinality, or a natural number. Typical facts include that every subset of a finite set is finite, and the union of finitely many finite sets is finite. Finite structures also arise across algebra, computer science, and logic: a finite group has a finite number of elements; a finite automaton has a finite number of states; a finite field contains a finite number of elements.

The term finites is usually encountered in informal discourse or in context where English grammar treats finite as a countable noun. In rigorous or published work, authors tend to specify the type of object (set, group, automaton, graph, field) rather than using the noun form "finites." When discussing finiteness, mathematicians emphasize the property of having a finite cardinality or size, as opposed to infiniteness, which leads to different theoretical and computational behaviors.

See also finite set, finite group, finite automaton, finite field, finite structure.