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Algebraic structure with only one element
In algebra, the zero object of a given algebraic structure is, in the sense explained below, the simplest object of such structure. As a set it is a singleton
Zero_object_(algebra)
Special objects used in (mathematical) category theory
of vector spaces over a field. See Zero object (algebra) for details. This is the origin of the term "zero object". In Ring, the category of rings with
Initial_and_terminal_objects
Unique ring consisting of one element
of rings, the zero ring is the terminal object, whereas the ring of integers Z is the initial object. The zero ring, denoted {0} or simply 0, consists
Zero_ring
Generalizations of '"`UNIQ--math-00000046-QINU`"' in algebraic structures
In mathematics, a zero element is one of several generalizations of the number zero to other algebraic structures. These alternate meanings may or may
Zero_element
Topics referred to by the same term
is zero Zero (complex analysis), a zero of a holomorphic function Zero element, generalization of the number zero in algebraic structures Zero object (algebra)
Zero_(disambiguation)
Mathematical object studied in the field of algebraic geometry
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as
Algebraic_variety
Elements taken to zero by a homomorphism
Kernels allow defining quotient objects (also called quotient algebras in universal algebra). For many types of algebraic structure, the fundamental theorem
Kernel_(algebra)
Branch of mathematics
"tangible" mathematical objects. A spectral sequence is a powerful tool for this. It has played an enormous role in algebraic topology. Its influence
Homological_algebra
Theory of algebraic structures in general
the object of study—this is the subject of group theory and ring theory— in universal algebra, the object of study is the possible types of algebraic structures
Universal_algebra
Number
for the World's First Zero A History of Zero Zero Saga The History of Algebra Edsger W. Dijkstra: Why numbering should start at zero, EWD831 (PDF of a handwritten
0
Quality of zero being an even number
number of objects is even. If an object is left over, then the number of objects is odd. The empty set contains zero groups of two, and no object is left
Parity_of_zero
Algebraic structure with addition and multiplication
In mathematics, a ring is an algebraic structure consisting of a set with two binary operations typically called addition and multiplication and denoted
Ring_(mathematics)
Group that has only one element
the trivial non-strict order ≤ {\displaystyle \,\leq } . Zero object (algebra) – Algebraic structure with only one element List of small groups Rowland
Trivial_group
Direct sum of simple Lie algebras
Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper
Semisimple_Lie_algebra
Mathematical object
In mathematics, an initial algebra is an initial object in the category of F-algebras for a given endofunctor F. This initiality provides a general framework
Initial_algebra
Construction of a ring of fractions
algebraic geometry: if R is a ring of functions defined on some geometric object (algebraic variety) V, and one wants to study this variety "locally" near a point
Localization (commutative algebra)
Localization_(commutative_algebra)
Branch of mathematics
set of these solutions. Abstract algebra studies algebraic structures, which consist of a set of mathematical objects together with one or several operations
Algebra
Algebraic structure used in analysis
and classification of Lie groups in terms of Lie algebras, which are simpler objects of linear algebra. In more detail: for any Lie group, the multiplication
Lie_algebra
Algebraic structure with addition, multiplication, and division
functions on X. The function field of an algebraic variety X (a geometric object defined as the common zeros of polynomial equations) consists of ratios
Field_(mathematics)
Set with operations obeying given axioms
universal algebra, an algebraic structure is called an algebra; this term may be ambiguous, since, in other contexts, an algebra is an algebraic structure
Algebraic_structure
Smallest integer n for which n equals 0 in a ring
numbers R {\displaystyle \mathbb {R} } and all algebraic number fields. Other fields of characteristic zero are the p-adic fields that are widely used in
Characteristic_(algebra)
Function type in category theory
F(A)\rightarrow A} . The object A {\displaystyle A} is called the carrier of the algebra. When it is permissible from context, algebras are often referred to
F-algebra
Branch of mathematics
Linear algebra is the branch of mathematics concerning linear equations such as a 1 x 1 + ⋯ + a n x n = b , {\displaystyle a_{1}x_{1}+\cdots +a_{n}x_{n}=b
Linear_algebra
Algebra used in 2D conformal field theories and string theory
In mathematics, a vertex operator algebra (VOA) is an algebraic structure that plays an important role in two-dimensional conformal field theory and string
Vertex_operator_algebra
Algebraic structure designed for geometry
geometric algebra (also known as a Clifford algebra) is an algebra that can represent and manipulate geometrical objects such as vectors. Geometric algebra is
Geometric_algebra
Number in {..., –2, –1, 0, 1, 2, ...}
is a commutative ring with unity. It is the prototype of all objects of such algebraic structure. Only those equalities of expressions are true in
Integer
Exact sequence used to describe the structure of an object
algebra, a resolution (or left resolution; dually a coresolution or right resolution) is an exact sequence of modules (or, more generally, of objects
Resolution_(algebra)
Category whose objects are rings and whose morphisms are ring homomorphisms
The action of a monoid (= commutative ring) R on an object (= ring) A of Ring is an R-algebra. The category of rings has a number of important subcategories
Category_of_rings
Branch of mathematics
studies zeros of multivariate polynomials; the modern approach generalizes this in a few different aspects. The fundamental objects of study in algebraic geometry
Algebraic_geometry
known as the group algebra; it is an R-module equipped with a multiplication. A group is the same as a category with a single object in which all morphisms
Category_algebra
Algebraic structure with "nice" duality properties
theory, a Frobenius algebra is a finite-dimensional unital associative algebra with a special kind of bilinear form which gives the algebras particularly nice
Frobenius_algebra
Scientific area at the interface between computer science and mathematics
manipulating mathematical expressions and other mathematical objects. Although computer algebra could be considered a subfield of scientific computing, they
Computer_algebra
Free object in the category of associative algebras
category of R-algebras to the category of sets. Free algebras over division rings are free ideal rings. Cofree coalgebra Tensor algebra Free object Noncommutative
Free_algebra
Mathematical expression using basic operations
numbers, any algebraic expression can be called an arithmetic expression. However, algebraic expressions can be used on more abstract objects such as in
Algebraic_expression
Algebra can essentially be considered as doing computations similar to those of arithmetic but with non-numerical mathematical objects. However, until
History_of_algebra
Type of geometric algebra
Conformal geometric algebra (CGA) is the geometric algebra constructed over the resultant space of a map from points in an n-dimensional base space Rp
Conformal_geometric_algebra
Ring that is also a vector space or a module
unital associative R-algebra is a monoid object in R-Mod (the monoidal category of R-modules). By definition, a ring is a monoid object in the category of
Associative_algebra
Algebraic structure with a binary operation
In abstract algebra, a magma, binar, or, rarely, groupoid is a basic kind of algebraic structure. Specifically, a magma consists of a set equipped with
Magma_(algebra)
Topics referred to by the same term
C* is an object-oriented programming language. It may also refer to: Apache Cassandra, a database system C*-algebra, an algebra Star and crescent, the
C*_(disambiguation)
Mathematical ring whose elements are matrices
In abstract algebra, a matrix ring is a set of matrices with entries in a ring R that form a ring under matrix addition and matrix multiplication. The
Matrix_ring
Construction in algebra
In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously a (unital associative) algebra and a (counital coassociative)
Hopf_algebra
Type of algebras, possibly non associative
norm of the algebra. A composition algebra (A, ∗, N) is either a division algebra or a split algebra, depending on the existence of a non-zero v in A such
Composition_algebra
Algebra associated to any vector space
In mathematics, the exterior algebra or Grassmann algebra of a vector space V {\displaystyle V} is an associative algebra that contains V , {\displaystyle
Exterior_algebra
Square matrix with ones on the main diagonal and zeros elsewhere
diagonal and zeros elsewhere. It has unique properties; for example when the identity matrix represents a geometric transformation, the object remains unchanged
Identity_matrix
Type of mathematical expression
takes the value zero are generally called zeros instead of "roots". The study of the sets of zeros of polynomials is the object of algebraic geometry. For
Polynomial
Branch of mathematics that studies abstract algebraic structures
representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations (for example, matrix
Representation_theory
"Smallest" commutative algebra that contains a vector space
mathematics, the symmetric algebra S(V) (also denoted Sym(V)) on a vector space V over a field K is a commutative algebra over K that contains V, and
Symmetric_algebra
Measure of a mathematical object studied in the field of algebraic geometry
K be a field, and L ⊇ K be an algebraically closed extension. An affine algebraic set V is the set of the common zeros in Ln of the elements of an ideal
Dimension of an algebraic variety
Dimension_of_an_algebraic_variety
Function in algebra
generalizes to commutative algebra the notion of size inherent in consideration of the degree of a pole or multiplicity of a zero in complex analysis, the
Valuation_(algebra)
*-algebra of bounded operators on a Hilbert space
In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology
Von_Neumann_algebra
Algebraic structure with an associative operation and an identity element
In abstract algebra, a monoid is a set equipped with an associative binary operation and an identity element. For example, the natural numbers with addition
Monoid
Homomorphism from an initial algebra into another algebra
initial object of the Maybe-Algebra is the set of all objects of natural number type Nat together with the morphism ini defined below: data Nat = Zero | Succ
Catamorphism
In algebraic topology, an S {\displaystyle \mathbb {S} } -object (also called a symmetric sequence) is a sequence { X ( n ) } {\displaystyle \{X(n)\}}
S-object
Algebraic structure
The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not specific
Commutative_ring
Application of Clifford algebra
transformations and geometric objects out of them. Formally: it identifies planar reflections with the grade-1 elements of a Clifford Algebra, that is, elements
Plane-based_geometric_algebra
Standard representation of a mathematical object
computer algebra, when representing mathematical objects in a computer, there are usually many different ways to represent the same object. In this context
Canonical_form
Algebraic study of differential equations
algebraic objects in view of deriving properties of differential equations and operators without computing the solutions, similarly as polynomial algebras are
Differential_algebra
Theoretical object in mathematics
is a field-like object whose characteristic is one. Most proposed theories of F1 replace abstract algebra entirely. Mathematical objects such as vector
Field_with_one_element
Branch of number theory
Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields
Algebraic_number_theory
Branch of mathematics
properties of formal duals of non-commutative algebraic objects such as rings as well as geometric objects derived from them (e.g. by gluing along localizations
Noncommutative algebraic geometry
Noncommutative_algebraic_geometry
Boolean algebra with all operators and laws forming a complete logical system
a complete Boolean algebra is a Boolean algebra in which every subset has a supremum (least upper bound). Complete Boolean algebras are used to construct
Complete_Boolean_algebra
Structure-preserving function between two rings
is a terminal object in the category of rings. As the initial object is not isomorphic to the terminal object, there is no zero object in the category
Ring_homomorphism
Overview of and topical guide to category theory
Initial object Terminal object Zero object Subobject Group object Magma object Natural number object Exponential object Epimorphism Monomorphism Zero morphism
Outline_of_category_theory
Number property of being positive or negative
positive and a negative zero. In mathematics and physics, the phrase "change of sign" is associated with exchanging an object for its additive inverse
Sign_(mathematics)
Algebraic structure in homological algebra
homological algebra, algebraic topology, and algebraic geometry – a differential graded algebra (or DGA, or DG algebra) is an algebraic structure often
Differential_graded_algebra
Subject area in mathematics
algebraic, and arithmetic objects are assigned objects called K-groups. These are groups in the sense of abstract algebra. They contain detailed information
Algebraic_K-theory
Concept in mathematics
enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra. Universal
Universal_enveloping_algebra
Object that is both a product and coproduct
given by the disjoint union. This category does not have a zero object. Block matrix algebra relies upon biproducts in categories of matrices. If the biproduct
Biproduct
Algebraic manipulation of "true" and "false"
mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables
Boolean_algebra
mathematics, a stable ∞-category is an ∞-category such that (i) It has a zero object. (ii) Every morphism in it admits a fiber and cofiber. (iii) A triangle
Stable_∞-category
This glossary of linear algebra is a list of definitions and terms relevant to the field of linear algebra, the branch of mathematics concerned with linear
Glossary_of_linear_algebra
Branch of mathematics
differentiable. Algebraic geometry studies algebraic curves, which are defined as algebraic varieties of dimension one. A surface is a two-dimensional object, such
Geometry
Endomorphism algebra of an abelian group
initial object in the category of rings. In a similar fashion, if R is any commutative ring, the endomorphisms of an R-module form an algebra over R by
Endomorphism_ring
Idempotent linear transformation from a vector space to itself
In linear algebra and functional analysis, a projection is a linear transformation P {\displaystyle P} from a vector space to itself (an endomorphism)
Projection_(linear_algebra)
Extension of a mathematical field with polynomial roots
every element of L is a root of a non-zero polynomial with coefficients in K. A field extension that is not algebraic, is said to be transcendental, and
Algebraic_extension
Mathematical idealization of the trace left by a moving point
curve. A plane algebraic curve is the zero set of a polynomial in two indeterminates. More generally, an algebraic curve is the zero set of a finite
Curve
Algebraic structure in linear algebra
advanced abstract algebra—to indirectly define objects by specifying maps from or to this object. From the point of view of linear algebra, vector spaces
Vector_space
Quotient space of a codomain of a linear map by the map's image
operator between Hilbert spaces) is an object Q and a morphism q : Y → Q such that the composition q f is the zero morphism of the category, and furthermore
Cokernel
Algebraic structure
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more
Polynomial_ring
Associative algebra used in combinatorics
prescribed zero-pattern determined by the incomparable elements in S under ≤. The incidence algebra of ≤ is then isomorphic to the algebra of upper-triangular
Incidence_algebra
Equivalence relation in algebra
that a congruence is an equivalence relation on an algebraic object that is compatible with the algebraic structure, in the sense that the operations are
Congruence_relation
Category with direct sums and certain types of kernels and cokernels
"abelian category". A category is abelian if it is preadditive and it has a zero object, it has all binary biproducts, it has all kernels and cokernels, and
Abelian_category
Generalization of vector spaces from fields to rings
central notions of commutative algebra and homological algebra, and are used widely in algebraic geometry and algebraic topology. In a vector space, the
Module_(mathematics)
Type of category in category theory
most general context in which the algebra of matrices makes sense. Recall that the morphisms from a single object A to itself form the endomorphism ring
Additive_category
Branch of mathematics that studies algebraic structures
algebra in Wiktionary, the free dictionary. In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures
List of abstract algebra topics
List_of_abstract_algebra_topics
Field of knowledge
scope of algebra thus grew to include the study of algebraic structures. This object of algebra was called modern algebra or abstract algebra, as established
Mathematics
Setting of relativistic physics in geometric algebra
spacetime algebra (STA) is the application of Clifford algebra Cl1,3(R), or equivalently the geometric algebra G(M4) of physics. Spacetime algebra provides
Spacetime_algebra
Map (arrow) between two objects of a category
that applies also to algebraic number theory. A category C {\displaystyle {\mathcal {C}}} consists of two classes, one of objects and the other of morphisms
Morphism
Special case of colimit in category theory
definition for algebraic structures like groups and modules, and then the general definition, which can be used in any category. In this section objects are understood
Direct_limit
Construction in category theory
of universal algebra, that is, a type of algebraic structures, whose axioms are unconditional (fields do not form an algebra, since zero does not have
Inverse_limit
Four-dimensional number system
division algebra over the real numbers. The next extension gives the sedenions, which have zero divisors and so cannot be a normed division algebra. The unit
Quaternion
Array of numbers
"two-by-three matrix", a 2 × 3 matrix, or a matrix of dimension 2 × 3. In linear algebra, matrices are used as linear maps. In geometry, matrices are used for geometric
Matrix_(mathematics)
Directed graph which is also a multigraph
the second, their product is defined to be zero. This defines an associative algebra over K. This algebra has a unit element if and only if the quiver
Quiver_(mathematics)
Reduction of a ring by one of its ideals
In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite
Quotient_ring
General theory of mathematical structures
work on algebraic topology. Category theory can be used in most areas of mathematics. In particular, many constructions of new mathematical objects from
Category_theory
Arithmetic operation
certain mathematical structures, division by zero is possible, such as in the zero ring and in algebraic structures such as wheels. In these structures
Division_(mathematics)
projective dimension of A as an (Aop ⊗R A)-module. For example, an algebra has bidimension zero if and only if it is separable. boolean A boolean ring is a ring
Glossary_of_ring_theory
Vector space consisting of affine subsets
In linear algebra, the quotient of a vector space V {\displaystyle V} by a subspace U {\displaystyle U} is a vector space obtained by "collapsing" U {\displaystyle
Quotient space (linear algebra)
Quotient_space_(linear_algebra)
German mathematician (1882–1935)
condition, and objects satisfying it are named Noetherian in her honor. In the third epoch (1927–1935), she published works on noncommutative algebras and hypercomplex
Emmy_Noether
Concept in mathematics
finite-dimensional Lie algebras is an equivalence of categories. Over fields of non-zero characteristic, formal group laws are not equivalent to Lie algebras. In fact
Formal_group_law
ZERO OBJECT-ALGEBRA
ZERO OBJECT-ALGEBRA
Boy/Male
Hindu, Indian
A Holy Object
Boy/Male
Arabic
Desire; Object
Girl/Female
Latin Greek Shakespearean
Daughter of Priam.
Male
Finnish
Finnish form of German Erich, EERO means "ever-ruler."Â
Boy/Male
Arabic, Australian, German, Greek, Kurdish
Empty; Void
Male
Spanish
Spanish name derived from Latin juniperus, JUNÃPERO means "juniper tree."
Boy/Male
Muslim
Desire. Object.
Boy/Male
Biblical
Root, that straitens or binds, that keeps tight.
Girl/Female
Arabic, Muslim
Rarity; Rare Object; Novelty
Male
Italian
 Short form of Italian Raniero, NERO means "wise warrior." Compare with another form of Nero.
Male
Finnish
Short form of Finnish Antero, TERO means "man; warrior."
Female
Greek
(ἩÏá½¼) Greek name derived form the word hÄ“rÅs, HERO means "hero." In mythology, this is the name of the lover of Leandros (Latin Leander).
Girl/Female
Gaelic Irish
Pointed object.
Boy/Male
Greek
Rock.
Boy/Male
Australian, Gaelic
Pointed Object
Boy/Male
Arabic
Empty.
Girl/Female
Muslim
Rarity, Rare object, Novelty
Girl/Female
Bengali, Indian
A Discovered Object
Girl/Female
Latin
Mother of Asopus.
Boy/Male
Indian, Sanskrit
God; Object of Worship
ZERO OBJECT-ALGEBRA
ZERO OBJECT-ALGEBRA
Boy/Male
Tamil
Enough
Boy/Male
Arabic
Happy; Pleased
Boy/Male
Hindu, Indian
Victory
Boy/Male
British, English
From the Welshman's Bridge
Boy/Male
Celtic
From the thorn bush or thicket.
Boy/Male
Hindu
Name of Shiva
Surname or Lastname
French and English
French and English : topographic name for someone who lived by a fortified stronghold, Old French, Middle English motte. The surname may also be a habitational name from any of the places in France named with this word.English : variant spelling of Mott 2.German : habitational name from Motte in the Saarland or Motten in Bavaria.The settlement that became the city of Detroit was founded in 1701 by Antoine de la Mothe, Sieur de Cadillac (1658–1730), governor of LA. He was born into the minor nobility in Gascony, France, where his father owned the seigneury of Cadillac.
Male
Hebrew
Variant spelling of Hebrew Yoqtan, YOKTAN means "small."Â
Female
Egyptian
, a goddess who was worshipped at Syene and Eilethya.
Boy/Male
British, English, Irish
Battle
ZERO OBJECT-ALGEBRA
ZERO OBJECT-ALGEBRA
ZERO OBJECT-ALGEBRA
ZERO OBJECT-ALGEBRA
ZERO OBJECT-ALGEBRA
a.
The person who is treated of; the hero of a piece; the chief character.
n.
A cipher; nothing; naught.
pl.
of Zero
a.
Sunk to a law condition; down in spirit or hope; degraded; servile; groveling; despicable; as, abject posture, fortune, thoughts.
imp. & p. p.
of Object
a.
Exposed; liable; prone; disposed; as, a country subject to extreme heat; men subject to temptation.
v. t.
To throw in; to dart in; to force in; as, to inject cold water into a condenser; to inject a medicinal liquid into a cavity of the body; to inject morphine with a hypodermic syringe.
pl.
of Zero
v. t.
That which is put, or which may be regarded as put, in the way of some of the senses; something visible or tangible; as, he observed an object in the distance; all the objects in sight; he touched a strange object in the dark.
n.
The point from which the graduation of a scale, as of a thermometer, commences.
n.
The common cero; also, the spotted cero. See Cero.
object.
Originally, an interrogative pronoun, later, a relative pronoun also; -- used always substantively, and either as singular or plural. See the Note under What, pron., 1. As interrogative pronouns, who and whom ask the question: What or which person or persons? Who and whom, as relative pronouns (in the sense of that), are properly used of persons (corresponding to which, as applied to things), but are sometimes, less properly and now rarely, used of animals, plants, etc. Who and whom, as compound relatives, are also used especially of persons, meaning the person that; the persons that; the one that; whosoever.
v. t.
To cause to undergo; as, to subject a substance to a white heat; to subject a person to a rigid test.
n.
Fig.: The lowest point; the point of exhaustion; as, his patience had nearly reached zero.
n.
A large and valuable fish of the Mackerel family, of the genus Scomberomorus. Two species are found in the West Indies and less commonly on the Atlantic coast of the United States, -- the common cero (Scomberomorus caballa), called also kingfish, and spotted, or king, cero (S. regalis).
n.
One who objects; one who offers objections to a proposition or measure.
v. t.
That which is set, or which may be regarded as set, before the mind so as to be apprehended or known; that of which the mind by any of its activities takes cognizance, whether a thing external in space or a conception formed by the mind itself; as, an object of knowledge, wonder, fear, thought, study, etc.
v. t.
A word, phrase, or clause toward which an action is directed, or is considered to be directed; as, the object of a transitive verb.
object.
The nominative case of the pronoun of the first person; the word with which a speaker or writer denotes himself.
n.
A cipher; zero.