endlichen

Endlichen is the inflected form of the German adjective endlich, meaning finite or limited. It is used to describe things that have a bound or are restricted in extent, duration, or quantity. In written German, endlichen typically appears in plural or in certain grammatical cases, for example in die endlichen Mengen (the finite sets) or mit den endlichen Automaten (with the finite automata).

In mathematics and related fields, endliche describes objects with a finite cardinality or duration. Common usages include endliche Mengen (finite sets), endliche Gruppen (finite groups), endliche Körper (finite fields), and endliche Folgen (finite sequences). These contrasts with unendliche (infinite) concepts, such as unendliche Mengen or unendliche Reihen.

In computer science and formal language theory, endliche Automaten (finite automata) and endliche Sprachen (finite languages) are standard terms. A endlicher Automat is a computational model with a finite set of states; it recognizes regular languages, some of which may be finite. The notion of Endlichkeit is central to questions about decidability and computational resources, particularly when comparing finite versus infinite structures.

Etymologically, endlichen derives from endlich (finite) and is used across disciplines to denote boundedness. The counterpart of endlichen is unendlich, meaning infinite. Related terms include Endlichkeit (the property of being finite) and Finite Automata in English-language literature.

See also: Endlichkeit, Unendlichkeit, Finite Automata, finite sets.