Word.ws English Dictionary

finitely generated

finitely generated adjective: such that the functor operatorname Hom_( mathcal )C(X,·) preserves those filtered colimits of monomorphisms.

Adjective

finitely generated (not comparable)

  1. (most generally) Such that the functor operatorname Hom_( mathcal )C(X,·) preserves those filtered colimits of monomorphisms.
  2. (less generally, but still covering most cases outside pure category theory) Being 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. (abstract, usually, module, ring) Having 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. (ring theory) Finitely generated as a (left) module over R.
  5. Equipped with an Alexandrov topology (i.e. one where the intersection of every family of open sets is open).

Origin of finitely generated

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.

Questions about finitely generated

What does finitely generated mean?
Finitely generated means such that the functor operatorname Hom_( mathcal )C(X,·) preserves those filtered colimits of monomorphisms.
What part of speech is finitely generated?
Finitely generated is an adjective.

Words near finitely generated in the dictionary

  1. finite-state automaton
  2. finite-state machine
  3. finite-state machines
  4. finiteless
  5. finitely
  6. finitely generated
  7. finitely presented
  8. finiteness
  9. finitenesses
  10. finites
  11. finitesimal

Source and license

Adapted from Wiktionary: finitely generated (page history and authors). Text available under the Creative Commons Attribution-ShareAlike 4.0 license. Word.ws reformatted the entry and added links. Wiktionary data of 2026-09-02.

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