finitely generated

adjective

Such that the functor operatorname Hom_( mathcal )C(X,·) preserves those filtered colimits of monomorphisms.

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Meanings

adjectivefinitely generated (not comparable)
  1. In any of several specific senses, such that all its elements can be created using (or described by reference to) a finite set of elements, usually called generators:
    1. not comparablemost generallyalgebraSuch that the functor operatorname Hom_( mathcal )C(X,·) preserves those filtered colimits of monomorphisms.
    2. not comparableless generally; but still covering most cases outside pure category theory; less generally; but still covering most cases outside pure category theoryalgebraBeing a quotient object of a free object over a finite set, i.e. being the target of a regular epimorphism from an object which is free on a finite set.
    3. abstractnot comparableusuallymodule; ring; module; ringalgebraHaving a finite set of generators, i.e. having a finite set of elements from which all other elements can be created in finitely many steps under the permitted operations (viz. the group operation for groups, addition and scalar multiplication for modules, addition and multiplication for rings, etc.)
    4. not comparablering theoryalgebraFinitely generated as a (left) module over R.
    5. not comparablealgebraEquipped with an Alexandrov topology (i.e. one where the intersection of every family of open sets is open).

Origin

From the study of generators. The motivation for calling topologies satisfying sense 1.5 "finitely generated" is that any topology satisfies sense 1.5 if and only if it is coherent with its finite subspaces. Thus, metaphorically, it is "generated" by them. The category-theoretic senses were created to generalize those of abstract algebra, and so were named identically.

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Cite this entry
APA

finitely generated. (n.d.). In WatkinsX Dictionary. Retrieved November 3, 2026, from https://dictionary.watkinsx.com/word/finitely%20generated

MLA

“finitely generated.” WatkinsX Dictionary, https://dictionary.watkinsx.com/word/finitely%20generated. Accessed November 3, 2026.

Chicago

WatkinsX Dictionary, s.v. “finitely generated,” accessed November 3, 2026, https://dictionary.watkinsx.com/word/finitely%20generated.

Source: finitely generated on Wiktionary, CC BY-SA 4.0.