Multiplicative group
In mathematics and group theory, the term multiplicative group refers to one of the following concepts:
* the group under multiplication of the invertible elements of a field, ring, or other structur
Cancellative semigroup
In mathematics, a cancellative semigroup (also called a cancellation semigroup) is a semigroup having the cancellation property. In intuitive terms, the cancellation property asserts that from an equa
Monus
In mathematics, monus is an operator on certain commutative monoids that are not groups. A commutative monoid on which a monus operator is defined is called a commutative monoid with monus, or CMM. Th
N-ary group
In mathematics, and in particular universal algebra, the concept of an n-ary group (also called n-group or multiary group) is a generalization of the concept of a group to a set G with an n-ary operat
Empty semigroup
In mathematics, a semigroup with no elements (the empty semigroup) is a semigroup in which the underlying set is the empty set. Many authors do not admit the existence of such a semigroup. For them a
Outline of algebraic structures
In mathematics, there are many types of algebraic structures which are studied. Abstract algebra is primarily the study of specific algebraic structures and their properties. Algebraic structures may
Lindenbaum–Tarski algebra
In mathematical logic, the Lindenbaum–Tarski algebra (or Lindenbaum algebra) of a logical theory T consists of the equivalence classes of sentences of the theory (i.e., the quotient, under the equival
Near-ring
In mathematics, a near-ring (also near ring or nearring) is an algebraic structure similar to a ring but satisfying fewer axioms. Near-rings arise naturally from functions on groups.
Rational monoid
In mathematics, a rational monoid is a monoid, an algebraic structure, for which each element can be represented in a "normal form" that can be computed by a finite transducer: multiplication in such
Numerical semigroup
In mathematics, a numerical semigroup is a special kind of a semigroup. Its underlying set is the set of all nonnegative integers except a finite number and the binary operation is the operation of ad
Torsion-free abelian group
In mathematics, specifically in abstract algebra, a torsion-free abelian group is an abelian group which has no non-trivial torsion elements; that is, a group in which the group operation is commutati
Primitive ring
In the branch of abstract algebra known as ring theory, a left primitive ring is a ring which has a faithful simple left module. Well known examples include endomorphism rings of vector spaces and Wey
Essential dimension
In mathematics, essential dimension is an invariant defined for certain algebraic structures such as algebraic groups and quadratic forms. It was introduced by J. Buhler and Z. Reichsteinand in its mo
Baumslag–Gersten group
In the mathematical subject of geometric group theory, the Baumslag–Gersten group, also known as the Baumslag group, is a particular one-relator group exhibiting some remarkable properties regarding i
Pseudo-ring
In mathematics, and more specifically in abstract algebra, a pseudo-ring is one of the following variants of a ring:
* A rng, i.e., a structure satisfying all the axioms of a ring except for the exis
Trivial semigroup
In mathematics, a trivial semigroup (a semigroup with one element) is a semigroup for which the cardinality of the underlying set is one. The number of distinct nonisomorphic semigroups with one eleme
Clifford semigroup
A Clifford semigroup (sometimes also called "inverse Clifford semigroup") is a completely regular inverse semigroup.It is an inverse semigroup with. Examples of Clifford semigroups are groups and comm
Special classes of semigroups
In mathematics, a semigroup is a nonempty set together with an associative binary operation. A special class of semigroups is a class of semigroups satisfying additional properties or conditions. Thus
E-semigroup
In the area of mathematics known as semigroup theory, an E-semigroup is a semigroup in which the idempotents form a subsemigroup. Certain classes of E-semigroups have been studied long before the more
Inverse semigroup
In group theory, an inverse semigroup (occasionally called an inversion semigroup) S is a semigroup in which every element x in S has a unique inverse y in S in the sense that x = xyx and y = yxy, i.e
Semigroup with involution
In mathematics, particularly in abstract algebra, a semigroup with involution or a *-semigroup is a semigroup equipped with an involutive anti-automorphism, which—roughly speaking—brings it closer to
Hardy field
In mathematics, a Hardy field is a field consisting of germs of real-valued functions at infinity that are closed under differentiation. They are named after the English mathematician G. H. Hardy.
Finitely generated abelian group
In abstract algebra, an abelian group is called finitely generated if there exist finitely many elements in such that every in can be written in the form for some integers . In this case, we say that
Domain (ring theory)
In algebra, a domain is a nonzero ring in which ab = 0 implies a = 0 or b = 0. (Sometimes such a ring is said to "have the zero-product property".) Equivalently, a domain is a ring in which 0 is the o
Module (mathematics)
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of module generalizes also the notion of abelian group, sinc
Matrix field
In abstract algebra, a matrix field is a field with matrices as elements. In field theory there are two types of fields: finite fields and infinite fields. There are several examples of matrix fields
Finite lattice representation problem
In mathematics, the finite lattice representation problem, or finite congruence lattice problem, asks whether every finite lattice is isomorphic to the congruence lattice of some finite algebra.
Semilattice
In mathematics, a join-semilattice (or upper semilattice) is a partially ordered set that has a join (a least upper bound) for any nonempty finite subset. Dually, a meet-semilattice (or lower semilatt
Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3.
Near-field (mathematics)
In mathematics, a near-field is an algebraic structure similar to a division ring, except that it has only one of the two distributive laws. Alternatively, a near-field is a near-ring in which there i
Lattice (order)
A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements has a unique
Composition ring
In mathematics, a composition ring, introduced in, is a commutative ring (R, 0, +, −, ·), possibly without an identity 1 (see non-unital ring), together with an operation such that, for any three elem
Pseudogroup
In mathematics, a pseudogroup is a set of diffeomorphisms between open sets of a space, satisfying group-like and sheaf-like properties. It is a generalisation of the concept of a group, originating h
E-dense semigroup
In abstract algebra, an E-dense semigroup (also called an E-inversive semigroup) is a semigroup in which every element a has at least one weak inverse x, meaning that xax = x. The notion of weak inver
Matrix ring
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 set of all n × n matrices with entries in R is a
Interior algebra
In abstract algebra, an interior algebra is a certain type of algebraic structure that encodes the idea of the topological interior of a set. Interior algebras are to topology and the modal logic S4 w
Elliptic algebra
In algebra, an elliptic algebra is a certain regular algebra of a Gelfand–Kirillov dimension three (quantum polynomial ring in three variables) that corresponds to a cubic divisor in the projective sp
Completely regular semigroup
In mathematics, a completely regular semigroup is a semigroup in which every element is in some subgroup of the semigroup. The class of completely regular semigroups forms an important subclass of the
Planar ternary ring
In mathematics, an algebraic structure consisting of a non-empty set and a ternary mapping may be called a ternary system. A planar ternary ring (PTR) or ternary field is special type of ternary syste
Ordered exponential field
In mathematics, an ordered exponential field is an ordered field together with a function which generalises the idea of exponential functions on the ordered field of real numbers.
Semigroup with two elements
In mathematics, a semigroup with two elements is a semigroup for which the cardinality of the underlying set is two. There are exactly five distinct nonisomorphic semigroups having two elements:
* O2
Class of groups
A class of groups is a set theoretical collection of groups satisfying the property that if G is in the collection then every group isomorphic to G is also in the collection. This concept arose from t
Kleene algebra
In mathematics, a Kleene algebra (/ˈkleɪni/ KLAY-nee; named after Stephen Cole Kleene) is an idempotent (and thus partially ordered) semiring endowed with a closure operator. It generalizes the operat
Grothendieck group
In mathematics, the Grothendieck group, or group of differences, of a commutative monoid M is a certain abelian group. This abelian group is constructed from M in the most universal way, in the sense
Action algebra
In algebraic logic, an action algebra is an algebraic structure which is both a residuated semilattice and a Kleene algebra. It adds the star or reflexive transitive closure operation of the latter to
Biracks and biquandles
In mathematics, biquandles and biracks are sets with binary operations that generalize quandles and racks. Biquandles take, in the theory of virtual knots, the place that quandles occupy in the theory
Groupoid
In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can b
Complete Heyting algebra
In mathematics, especially in order theory, a complete Heyting algebra is a Heyting algebra that is complete as a lattice. Complete Heyting algebras are the objects of three different categories; the
Partial groupoid
In abstract algebra, a partial groupoid (also called halfgroupoid, pargoid, or partial magma) is a set endowed with a partial binary operation. A partial groupoid is a partial algebra.
Quantum groupoid
In mathematics, a quantum groupoid is any of a number of notions in noncommutative geometry analogous to the notion of groupoid. In usual geometry, the information of a groupoid can be contained in it
Simplicial commutative ring
In algebra, a simplicial commutative ring is a commutative monoid in the category of simplicial abelian groups, or, equivalently, a simplicial object in the category of commutative rings. If A is a si
Damm algorithm
In error detection, the Damm algorithm is a check digit algorithm that detects all single-digit errors and all adjacent transposition errors. It was presented by H. Michael Damm in 2004.
BCK algebra
In mathematics, BCI and BCK algebras are algebraic structures in universal algebra, which were introduced by Y. Imai, K. Iséki and S. Tanaka in 1966, that describe fragments of the propositional calcu
MV-algebra
In abstract algebra, a branch of pure mathematics, an MV-algebra is an algebraic structure with a binary operation , a unary operation , and the constant , satisfying certain axioms. MV-algebras are t
Magma (algebra)
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 a single binary operation that must be closed by
J-structure
In mathematics, a J-structure is an algebraic structure over a field related to a Jordan algebra. The concept was introduced by to develop a theory of Jordan algebras using linear algebraic groups and
Symmetric inverse semigroup
In abstract algebra, the set of all partial bijections on a set X (a.k.a. one-to-one partial transformations) forms an inverse semigroup, called the symmetric inverse semigroup (actually a monoid) on
Monoid
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid
Boolean algebra (structure)
In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operat
Commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra
Semiprimitive ring
In algebra, a semiprimitive ring or Jacobson semisimple ring or J-semisimple ring is a ring whose Jacobson radical is zero. This is a type of ring more general than a semisimple ring, but where simple
Algebraic structure
In mathematics, an algebraic structure consists of a nonempty set A (called the underlying set, carrier set or domain), a collection of operations on A (typically binary operations such as addition an
Jónsson–Tarski algebra
In mathematics, a Jónsson–Tarski algebra or Cantor algebra is an algebraic structure encoding a bijection from an infinite set X onto the product X×X. They were introduced by Bjarni Jónsson and Alfred
Biordered set
A biordered set (otherwise known as boset) is a mathematical object that occurs in the description of the structure of the set of idempotents in a semigroup. The set of idempotents in a semigroup is a
Kasch ring
In ring theory, a subfield of abstract algebra, a right Kasch ring is a ring R for which every simple right R module is isomorphic to a right ideal of R. Analogously the notion of a left Kasch ring is
Semimodule
In mathematics, a semimodule over a semiring R is like a module over a ring except that it is only a commutative monoid rather than an abelian group.
Semifield
In mathematics, a semifield is an algebraic structure with two binary operations, addition and multiplication, which is similar to a field, but with some axioms relaxed.
Group (mathematics)
In mathematics, a group is a set and an operation that combines any two elements of the set to produce a third element of the set, in such a way that the operation is associative, an identity element
Additive group
An additive group is a group of which the group operation is to be thought of as addition in some sense. It is usually abelian, and typically written using the symbol + for its binary operation. This
Monogenic semigroup
In mathematics, a monogenic semigroup is a semigroup generated by a single element. Monogenic semigroups are also called cyclic semigroups.
Integral element
In commutative algebra, an element b of a commutative ring B is said to be integral over A, a subring of B, if there are n ≥ 1 and aj in A such that That is to say, b is a root of a monic polynomial o
Semigroupoid
In mathematics, a semigroupoid (also called semicategory, naked category or precategory) is a partial algebra that satisfies the axioms for a small category, except possibly for the requirement that t
Pointed set
In mathematics, a pointed set (also based set or rooted set) is an ordered pair where is a set and is an element of called the base point, also spelled basepoint. Maps between pointed sets and – calle
Rng (algebra)
In mathematics, and more specifically in abstract algebra, a rng (or non-unital ring or pseudo-ring) is an algebraic structure satisfying the same properties as a ring, but without assuming the existe
Band (algebra)
In mathematics, a band (also called idempotent semigroup) is a semigroup in which every element is idempotent (in other words equal to its own square). Bands were first studied and named by A. H. Clif
Semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multip
Exponential field
In mathematics, an exponential field is a field that has an extra operation on its elements which extends the usual idea of exponentiation.
Moufang polygon
In mathematics, Moufang polygons are a generalization by Jacques Tits of the Moufang planes studied by Ruth Moufang, and are irreducible buildings of rank two that admit the action of .In a book on th
Nowhere commutative semigroup
In mathematics, a nowhere commutative semigroup is a semigroup S such that, for all a and b in S, if ab = ba then a = b. A semigroup S is nowhere commutative if and only if any two elements of S are i
Semigroup with three elements
In abstract algebra, a semigroup with three elements is an object consisting of three elements and an associative operation defined on them. The basic example would be the three integers 0, 1, and −1,
Variety of finite semigroups
In mathematics, and more precisely in semigroup theory, a variety of finite semigroups is a class of semigroups having some nice algebraic properties. Those classes can be defined in two distinct ways
Regular semigroup
In mathematics, a regular semigroup is a semigroup S in which every element is regular, i.e., for each element a in S there exists an element x in S such that axa = a. Regular semigroups are one of th
Effect algebra
Effect algebras are partial algebras which abstract the (partial) algebraic properties of events that can be observed in quantum mechanics. Structures equivalent to effect algebras were introduced by
Field (mathematics)
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a
Right group
In mathematics, a right group is an algebraic structure consisting of a set together with a binary operation that combines two elements into a third element while obeying the right group axioms. The r
BF-algebra
In mathematics, BF algebras are a class of algebraic structures arising out of a symmetric "Yin Yang" concept for Bipolar Fuzzy logic, the name was introduced by Andrzej Walendziak in 2007. The name c
Ring (mathematics)
In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a ring is a set equipped with
Near-semiring
In mathematics, a near-semiring (also seminearring) is an algebraic structure more general than a near-ring or a semiring. Near-semirings arise naturally from functions on monoids.
Integral closure of an ideal
In algebra, the integral closure of an ideal I of a commutative ring R, denoted by , is the set of all elements r in R that are integral over I: there exist such that It is similar to the integral clo
Double groupoid
In mathematics, especially in higher-dimensional algebra and homotopy theory, a double groupoid generalises the notion of groupoid and of category to a higher dimension.
Partial algebra
In abstract algebra, a partial algebra is a generalization of universal algebra to partial operations.
Quantum differential calculus
In quantum geometry or noncommutative geometry a quantum differential calculus or noncommutative differential structure on an algebra over a field means the specification of a space of differential fo
Semiring
In abstract algebra, a semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse. The term rig is also used occasionally—this or
Affine monoid
In abstract algebra, a branch of mathematics, an affine monoid is a commutative monoid that is finitely generated, and is isomorphic to a submonoid of a free abelian group . Affine monoids are closely
Epigroup
In abstract algebra, an epigroup is a semigroup in which every element has a power that belongs to a subgroup. Formally, for all x in a semigroup S, there exists a positive integer n and a subgroup G