finitely generated
adjEtymology 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.
› In any of several specific senses, such that…
Algebra, Category theory, Computing, Engineering, Mathematics, Natural sciences, Physical sciences, Sciences Such that the functor operatorname Hom_( mathcal )C(X,·) preserves those filtered colimits of monomorphisms.
Of an object X in a locally small category 𝒞 that admits filtered colimits of monomorphism
Algebra, Mathematics, Sciences 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.
Less generally; but still covering most cases outside pure category theory; less generally; but still covering most cases outside pure category theory Of an object X in a concrete category
Algebra, Mathematics, Sciences Abstract, Usually, Of a group-like structure 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.)
Module; ring; module; ring
Algebra, Mathematics, Sciences Finitely generated as a (left) module over R.
Of a (left) ideal in a ring R
Algebra, Mathematics, Sciences Of a space Equipped with an Alexandrov topology (i.e. one where the intersection of every family of open sets is open).