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Concept in category theory
In category theory, a finitely generated object is the quotient of a free object over a finite set, in the sense that it is the target of a regular epimorphism
Finitely_generated_object
Type of algebra
mathematics, a finitely generated algebra (also called an algebra of finite type) over a (commutative) ring R {\displaystyle R} , or a finitely generated R {\displaystyle
Finitely_generated_algebra
Topics referred to by the same term
scheme, a global version of a finitely presented algebra finitely presentable object, in category theory Finitely generated object This disambiguation page
Finitely_presented
In algebra, module with a finite generating set
concepts of finitely generated, finitely presented and coherent modules coincide. A finitely generated module over a field is simply a finite-dimensional
Finitely_generated_module
-presentable object is called finitely presentable. In the category Set of all sets, the finitely presentable objects coincide with the finite sets. The
Accessible_category
Type of Abelian category (in category theory in mathematics)
any non-zero finitely generated objects. A Grothendieck category is called locally finitely generated if it has a set of finitely generated generators (i
Grothendieck_category
Direct summand of a free module (mathematics)
However, it is true that for finitely presented modules M over a commutative ring R (in particular if M is a finitely generated R-module and R is Noetherian)
Projective_module
Algebraic structure with an associative operation and an identity element
to generate M if the smallest submonoid of M containing S is M. If there is a finite set that generates M, then M is said to be a finitely generated monoid
Monoid
Algebraic structure with only one element
module, is a finitely-generated module with an empty generating set. For structures requiring the multiplication structure inside the zero object, such as
Zero_object_(algebra)
Mathematical term in group theory
Grigorchuk group is a finitely generated group constructed by Rostislav Grigorchuk that provided the first example of a finitely generated group of intermediate
Grigorchuk_group
Mathematical concept
objects, also referred to as finitely presented objects, or objects of finite presentation, are objects in a category satisfying a certain finiteness
Compact_object_(mathematics)
Template that specifies one or more axioms
ZFC by replacing them with finitely many axioms in the same language. A theory is finitely axiomatizable if there is a finite set of sentences whose deductive
Axiom_schema
Construction of a larger algebraic field by "adding elements" to a smaller field
\ldots ,x_{n}\}),} and one says that K ( S ) {\displaystyle K(S)} is finitely generated over K {\displaystyle K} . If S {\displaystyle S} consists of a single
Field_extension
Mathematical group based upon a finite number of elements
Ludwig Stickelberger and later was both simplified and generalized to finitely generated modules over a principal ideal domain, forming an important chapter
Finite_group
Type of mathematical group
0 must be finite; Schur's theorem: a torsion linear group is locally finite. In particular, if it is finitely generated then it is finite. Selberg's
Linear_group
Special case of colimit in category theory
collection of finitely generated subgroups H i {\displaystyle H_{i}} of a given group G {\displaystyle G} can be partially ordered by inclusion. Finite sets of
Direct_limit
Mathematical concept
power set of a finite set is finite there can be only finitely many open sets (and only finitely many closed sets). A topology on a finite set can also
Finite_topological_space
Mathematical set closed under positive linear combinations
polyhedral if it is the conical hull of finitely many vectors (this property is also called finitely-generated). I.e., there is a set of vectors { v 1
Convex_cone
module M is dualizable if and only if it is a finitely generated projective module. In that case the dual object M∗ is also given by the module of homomorphisms
Dual_object
Mathematical study of invariants under symmetries
a finite group acting on a space over a field of characteristic zero, see Hilbert's theorems below. If the ring of invariants is finitely generated, then
Invariant_theory
Branch of mathematics that studies the properties of groups
finiteness assumptions are satisfied, for example finitely generated groups, or finitely presented groups (i.e. in addition the relations are finite)
Group_theory
Method in mathematical logic
a (countable) structure by its finitely generated substructures. Given a class K {\displaystyle \mathbf {K} } of finite relational structures, if K {\displaystyle
Fraïssé_limit
locally finite if every finitely generated subgroup is finite, and a group is locally soluble if every finitely generated subgroup is soluble. For finite groups
Local_property
Describes the objects of a given type, up to some equivalence
Classification of Finitely generated abelian group – Commutative group where every element is the sum of elements from one finite subset Classification
Classification_theorem
Smallest cardinality of a generating set for a group
rank(G)=0 is undecidable for finitely presented groups. The rank problem is decidable for finite groups and for finitely generated abelian groups. The rank
Rank_of_a_group
Formal power series
idea of generating functions can be extended to sequences of other objects. Thus, for example, polynomial sequences of binomial type are generated by: e
Generating_function
Numerical method for solving physical or engineering problems
implemented by the construction of a mesh of the object: the numerical domain for the solution that has a finite number of points. FEM formulation of a boundary
Finite_element_method
Groups of point isometries in 3 dimensions
bounded object is equal to its full symmetry group if and only if the object is chiral. The point groups that are generated purely by a finite set of reflection
Point groups in three dimensions
Point_groups_in_three_dimensions
is covered by finitely many affine open sets Spec A {\displaystyle {\text{Spec }}A} where each A {\displaystyle A} is finitely generated as a B {\displaystyle
Glossary of algebraic geometry
Glossary_of_algebraic_geometry
every object has finite length. This includes as a special case the category of finite-dimensional modules over an algebra. The category of finitely-generated
Krull–Schmidt_category
Discrete analog of a derivative
self-standing mathematical objects in works by George Boole (1860), L. M. Milne-Thomson (1933), and Károly Jordan [de] (1939). Finite differences trace their
Finite_difference
if and only if it is finitely generated and of finite projective dimension. Perfect complexes are precisely the compact objects in the unbounded derived
Perfect_complex
Mathematics concept
\varphi } can be generated by the conjugates of finitely many elements of F {\displaystyle F} , then G {\displaystyle G} is finitely presented. If S {\displaystyle
Free_group
Topic in abstract algebra
generalizations of this. Suppose that A is a finite-dimensional unital associative algebra over some field. A finitely-generated right A-module T is called a tilting
Tilting_theory
Branch of mathematics that studies algebraic structures
hull Flat module Flat cover Coherent module Finitely-generated module Finitely-presented module Finitely related module Algebraically compact module Reflexive
List of abstract algebra topics
List_of_abstract_algebra_topics
Category with direct sums and certain types of kernels and cokernels
category. The category of all finitely generated abelian groups is also an abelian category, as is the category of all finite abelian groups. If R is a ring
Abelian_category
Algebraic structure
any ideal is generated by finitely many elements, or, yet equivalent, submodules of finitely generated modules are finitely generated. Being Noetherian
Commutative_ring
for a finite group G acting on a finitely generated algebra R: since R is integral over RG, the Artin–Tate lemma implies RG is a finitely generated algebra
Fixed-point_subring
Area in mathematics devoted to the study of finitely generated groups
Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties
Geometric_group_theory
Exact sequence used to describe the structure of an object
However, a finite resolution is one where only finitely many of the objects in the sequence are non-zero; it is usually represented by a finite exact sequence
Resolution_(algebra)
Category-theoretic construction
product which have only finitely many nonzero terms. (It therefore coincides exactly with the direct product in the case of finitely many factors.) Given
Coproduct
Category whose objects are rings and whose morphisms are ring homomorphisms
reflective subcategory of CRing. The category of fields is neither finitely complete nor finitely cocomplete. In particular, Field has neither products nor coproducts
Category_of_rings
free. A module M over a ring R is stably free if there exists a free finitely generated module F over R such that M ⊕ F {\displaystyle M\oplus F} is a free
Stably_free_module
Set with associative invertible operation
the mid-1980s, geometric group theory, which studies finitely generated groups as geometric objects, has become an active area in group theory. One of the
Group_(mathematics)
underlying endofunctor commutes with filtered colimits. finite A category is finite if it has only finitely many morphisms. forgetful functor The forgetful functor
Glossary_of_category_theory
In mathematics, process for extending a category
ind-completion of the category FinSet of finite sets is the category of all sets. Similarly, if C is the category of finitely generated groups, ind-C is equivalent
Ind-completion
Submodule of a mathematical ring
see the general pattern for taking the sum of two finitely generated ideals, it is the ideal generated by the union of their generators. In the last three
Ideal_(ring_theory)
Generalization of vector spaces from fields to rings
of submodules becomes stationary after finitely many steps. Equivalently, every submodule is finitely generated. Artinian An Artinian module is a module
Module_(mathematics)
Group that admits a formal description in terms of reflections
and every triple edge by an edge labelled 6. Also note that every finitely generated Coxeter group is an automatic group. Dynkin diagrams have the additional
Coxeter_group
is finitely generated if there exist finitely many elements a1, ..., an such that I = a1R + ... + anR. A two-sided ideal I is finitely generated if there
Glossary_of_ring_theory
Property of topological spaces
Hausdorff. Compact-open topology – Type of topology Countably generated space Finitely generated space – Type of topology in mathematicsPages displaying short
Compactly_generated_space
Mathematical property
requiring that any finitely generated R-module M is semi-simple. Examples of semi-simple rings include fields and, more generally, finite direct products
Semi-simplicity
Deflection of a spinning object moving through a fluid
phenomenon that occurs when a spinning object is moving through a fluid. A lift force acts on the spinning object and its path may be deflected in a manner
Magnus_effect
an exceptional object is one of finitely many exceptions to some classification of objects. Many branches of mathematics study objects of a given type
Exceptional_object
In mathematics, a module that has a basis
f ( x ) = 0 for all but finitely many x ∈ E } . {\displaystyle R^{(E)}=\{f:E\to R\mid f(x)=0{\text{ for all but finitely many }}x\in E\}.} We equip
Free_module
Algebraic structure associated with a topological space
chain complex such that all but finitely many An are zero, and the others are finitely generated abelian groups (or finite-dimensional vector spaces), then
Homology_(mathematics)
Category of non-empty finite ordinals and order-preserving maps
{\displaystyle \Delta _{+}} is the monoidal category freely generated by a single monoid object, given by [ 0 ] {\displaystyle [0]} with the unique possible
Simplex_category
Roughly, the number of k-dimensional holes on a topological surface
finitely generated homology. Given a topological space which has a finitely generated homology, the Poincaré polynomial is defined as the generating function
Betti_number
Group of transformations under which the object is invariant
the groups of two elements generated by a reflection; they are isomorphic with C2 the infinite discrete groups generated by a translation; they are isomorphic
Symmetry_group
Concept in algebraic geometry
canonical ring is finitely generated, the canonical model is Proj of the canonical ring. If the canonical ring is not finitely generated, then Proj R is
Canonical_bundle
Type of plane partition
each of a given set of objects. It can be classified also as a tessellation. In the simplest case, these objects are just finitely many points in the plane
Voronoi_diagram
Mathematical element
f} is finite ( B {\displaystyle B} finitely generated A {\displaystyle A} -module) or of finite type ( B {\displaystyle B} finitely generated A {\displaystyle
Integral_element
Study of discrete mathematical structures
"discrete mathematics". The set of objects studied in discrete mathematics can be finite or infinite. The term finite mathematics is sometimes applied to
Discrete_mathematics
Categorical generalization of a function space in set theory
object or map object is the categorical generalization of a function space in set theory. Categories with all finite products and exponential objects
Exponential_object
Mathematical group that can be generated as the set of powers of a single element
group (meaning that its group operation is commutative), and every finitely generated abelian group is a direct product of cyclic groups. Every cyclic group
Cyclic_group
Branch of number theory
group of units of O, denoted O×, is a finitely generated abelian group. The fundamental theorem of finitely generated abelian groups therefore implies that
Algebraic_number_theory
Abelian group extending a commutative monoid
abelian group generated by symbols [ A ] {\displaystyle [A]} for any finitely generated abelian groups A. One first notes that any finite abelian group
Grothendieck_group
Commutative algebra theorem
theorem states that every finitely generated projective module over a polynomial ring is free. Geometrically, finitely generated projective modules over
Quillen–Suslin_theorem
F_{i}} are finitely generated free modules. 2. A finitely presented module is a module that admits a finite free presentation. finitely generated A module
Glossary_of_module_theory
Category admitting tensor products
category. Any category with finite products can be regarded as monoidal with the product as the monoidal product and the terminal object as the unit. Such a category
Monoidal_category
Type of countable group in group theory
a finitely presented group that contains a copy of every finitely presented group. If W {\displaystyle {\mathcal {W}}} is the class of all finitely presented
SQ-universal_group
Complex manifolds in mathematics
∗ ω Y ⊗ n {\displaystyle \oplus _{n}f_{*}\omega _{Y}^{\otimes n}} finitely generated? If this is true then P r o j ⊕ n f ∗ ω Y ⊗ n → X {\displaystyle Proj\oplus
Relative_canonical_model
Direct limit of a direct system of groups
collection of finitely generated subgroups H i {\displaystyle H_{i}} of a given group G {\displaystyle G} can be partially ordered by inclusion. Finite sets of
Direct_limit_of_groups
Injective polynomial functions are bijective
polynomial map of n {\displaystyle n} -dimensional affine space over a finitely generated extension of Z {\displaystyle \mathbb {Z} } or Z / p Z [ t ] {\displaystyle
Ax–Grothendieck_theorem
is infinite. Finitely-generated modules over principal ideal domains (PIDs) are classified by the structure theorem for finitely generated modules over
Indecomposable_module
Concept in probability theory
see real trees). This definition gives the finite-dimensional laws of the subtrees generated by finitely many leaves. Let us consider the space of all
Brownian_tree
Type of mathematical object
of copies of Gm, or as groups of multiplicative type associated to finitely generated free abelian groups. The general linear group GLn is an affine algebraic
Group_scheme
Mathematical object studied in the field of algebraic geometry
_{i=-\infty }^{\infty }A_{i}/{A_{i-1}}} is commutative, reduced and finitely generated as a k-algebra; i.e., gr A is the coordinate ring of an affine (reducible)
Algebraic_variety
Bounds of a sequence
elements which are in all except finitely many sets of the sequence and does not include elements which are in all except finitely many complements of sets of
Limit inferior and limit superior
Limit_inferior_and_limit_superior
Mathematical object in abstract algebra
the field k with finite dimension over k, then Homk(−, k) is a duality between finitely generated left A-modules and finitely generated right A-modules
Injective_module
Type of equation with integer coefficients
K {\displaystyle K} is the field generated by the roots of f {\displaystyle f} , then the equation has only finitely many solutions with x {\displaystyle
Thue_equation
mathematics, a coherent topos is a topos generated by a collection of quasi-compact quasi-separated objects closed under finite products. Deligne's completeness
Coherent_topos
Shock wave from flying at the speed of sound
with shock waves created when an object travels through the air faster than the speed of sound. Sonic booms generate enormous amounts of sound energy
Sonic_boom
Discrete subgroup in a locally compact topological group
is uniform and that lattices in solvable groups are finitely presented. Not all finitely generated solvable groups are lattices in a Lie group. An algebraic
Lattice_(discrete_subgroup)
Ukrainian mathematician
group is finite. Grigorchuk's group is a central object in the study of the so-called branch groups and automata groups. These are finitely generated groups
Rostislav_Grigorchuk
In algebra, completion w.r.t. powers of an ideal
topological module over R ^ I {\displaystyle {\widehat {R}}_{I}} if I is finitely generated. The ring of p-adic integers Z p {\displaystyle \mathbb {Z} _{p}}
Completion_of_a_ring
Algebraic structure with "nice" duality properties
k, a finite-dimensional, unital, associative k-algebra is a Frobenius algebra if it has only finitely many minimal right ideals. If F is a finite-dimensional
Frobenius_algebra
Braunling, Groechenig & Wolfson (2016) showed that Tate objects (for C the category of finitely generated projective R-modules, and subject to the condition
Tate_vector_space
Vertices connected in pairs by edges
regular examples of directed graphs are given by the Cayley graphs of finitely-generated groups, as well as Schreier coset graphs In category theory, every
Graph_(discrete_mathematics)
Standard representation of a mathematical object
decimal representation is a finite sequence of digits that does not begin with zero. More generally, for a class of objects on which an equivalence relation
Canonical_form
but it is finitely co-Hopfian. A finitely generated group G is called scale-invariant if there exists a nested sequence of subgroups of finite index of
Co-Hopfian_group
Planar movement within a Euclidean space without rotation
groups, but unlike the translation groups, are finitely generated. That is, a finite generating set generates the entire group. A translation is an affine
Translation_(geometry)
Subgroup of a root system's isometry group
subgroup which is generated by reflections through the hyperplanes orthogonal to at least one of the roots, and as such is a finite reflection group.
Weyl_group
Type of shift space studied in ergodic theory
of finite type are shift spaces defined by a finite set of forbidden words. They are used to model dynamical systems, and in particular are objects of
Subshift_of_finite_type
Class of algebraic structures
don't form a variety, as an arbitrary product of finitely generated abelian groups is not finitely generated. Viewing a variety V and its homomorphisms as
Variety_(universal_algebra)
Subject area in mathematics
Indeed, the global dimension of regular rings is finite, i.e. any finitely generated module has a finite projective resolution P* → M, and a simple argument
Algebraic_K-theory
Function that returns its argument unchanged
ISBN 978-3-319-31159-3. Rosales, J. C.; García-Sánchez, P. A. (1999). Finitely Generated Commutative Monoids. Nova Publishers. p. 1. ISBN 978-1-56072-670-8
Identity_function
Theorem classifying finite simple groups
the structure of finite groups (and their action on other mathematical objects) can sometimes be reduced to questions about finite simple groups. Thanks
Classification of finite simple groups
Classification_of_finite_simple_groups
Computational problems no algorithm can solve
entries generates a free semigroup. Determining whether two finitely generated subsemigroups of integer matrices have a common element. Given a finite set
List_of_undecidable_problems
Area of combinatorics that deals with the number of ways certain patterns can be formed
of finite sets Si indexed by the natural numbers, enumerative combinatorics seeks to describe a counting function which counts the number of objects in
Enumerative_combinatorics
FINITELY GENERATED-OBJECT
FINITELY GENERATED-OBJECT
Girl/Female
Tamil
Generates harmony in dance and music
Boy/Male
American, Australian, Christian, Danish, French, Gaelic, Latin, Scottish
Son of; Taken from Mackenzie; Greatest; Finely Made; Comely
Girl/Female
Indian
Beautiful, Virtuous, Venerated
Girl/Female
American, Australian, Chinese, Scottish
Son of the Fair One; Fair Skinned; Comely; Finely Made
Girl/Female
Tamil
Anindita | அநிஂதிதா
Beautiful, Virtuous, Venerated
Anindita | அநிஂதிதா
Girl/Female
Biblical
Penetrated.
Male
Gaelic
Variant form of Gaelic Cainneach, COINNEACH means "comely; finely made."
Girl/Female
African, American, British, Celtic, Chinese, Christian, English, Irish
Handsome; Beautiful; Knowing; Good Looking; Born of Fire; Finely Made
Boy/Male
Hindu, Indian
Self Generated; Lord Shiva
Boy/Male
Arabic, Muslim, Sindhi
Venerated; Honoured
Girl/Female
Hindu
Generates harmony in dance and music
Boy/Male
Indian, Telugu
Generated
Boy/Male
Indian
Honored, Venerated
Boy/Male
Muslim
Honored, Venerated
Girl/Female
Tamil
Aninditha | அநிஂதிதா
Beautiful, Virtuous, Venerated
Aninditha | அநிஂதிதா
Biblical
penetrated
Girl/Female
Assamese, Gujarati, Hindu, Indian, Kannada, Sindhi, Tamil, Telugu
Virtuous; Venerated
Boy/Male
Indian, Sanskrit
Devoted; Venerated
Girl/Female
Indian
Who is to be Venerated and Respected
Girl/Female
Indian
Beautiful, Virtuous, Venerated
FINITELY GENERATED-OBJECT
FINITELY GENERATED-OBJECT
Male
Greek
(ὙμÎναιος) Greek name HYMENAIOS means "bridal song" or "wedding song." In mythology, this is the name of a god of marriage.
Boy/Male
Muslim
Clear
Girl/Female
Celtic English
Strong. She ascends. Feminine of Brian.
Girl/Female
Assamese, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Telugu
Lake; Jolly; Happy
Boy/Male
British, Christian, English, Gaelic, Irish, Welsh
Sharp; Warrior's Son; Ancient; Beautiful; Distant
Girl/Female
Tamil
Initiation
Male
Russian
Variant spelling of Russian Dmitriy, DMITRI means "loves the earth" or "follower of Demeter."
Girl/Female
English American
Meadow of ash trees.
Surname or Lastname
English
English : habitational name from places so named in Cheshire, Derbyshire, Herefordshire, Leicestershire, Shropshire, Staffordshire, and Warwickshire. Compare Stratton.
Boy/Male
Hindu, Indian, Marathi
Risen; Elevated
FINITELY GENERATED-OBJECT
FINITELY GENERATED-OBJECT
FINITELY GENERATED-OBJECT
FINITELY GENERATED-OBJECT
FINITELY GENERATED-OBJECT
n.
That which generates.
a.
Having a limit; limited in quantity, degree, or capacity; bounded; -- opposed to infinite; as, finite number; finite existence; a finite being; a finite mind; finite duration.
n.
Capability of being generated.
adv.
Infinitely.
a.
Not begot; not yet generated; also, having never been generated; self-existent; eternal.
imp. & p. p.
of Venerate
a.
Capable of being generated or produced.
a.
Generated by water.
v. t.
To generate; to produce.
a.
Relating to autogenesis; self-generated.
p. pr. & vb. n.
of Generate
n.
One who, or that which, generates, begets, causes, or produces.
v. t.
To regard with reverential respect; to honor with mingled respect and awe; to reverence; to revere; as, we venerate parents and elders.
v. t.
To beget; to procreate; to propagate; to produce (a being similar to the parent); to engender; as, every animal generates its own species.
imp. & p. p.
of Generate
a.
Self-generated; produced independently.
adv.
Without bounds or limits; beyond or below assignable limits; as, an infinitely large or infinitely small quantity.
a.
First formed or generated; original; primigenial.
adv.
In a finite manner or degree.