Algebraic structure with an associative operation and an identity element
For monoid objects in category theory, see Monoid (category theory).
Not to be confused with Monad.
Algebraic structures
Group-like
Group
Semigroup / Monoid
Rack and quandle
Quasigroup and loop
Abelian group
Magma
Lie group
Group theory
Ring-like
Ring
Rng
Semiring
Near-ring
Commutative ring
Domain
Integral domain
Field
Division ring
Lie ring
Ring theory
Lattice-like
Lattice
Semilattice
Complemented lattice
Total order
Heyting algebra
Boolean algebra
Map of lattices
Lattice theory
Module-like
Module
Group with operators
Vector space
Linear algebra
Algebra-like
Algebra
Associative
Non-associative
Composition algebra
Lie algebra
Graded
Bialgebra
Hopf algebra
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Algebraic structures between magmas and groups. For example, monoids are semigroups with identity.
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, the identity element being 0.
Monoids are semigroups with identity. Such algebraic structures occur in several branches of mathematics.
The functions from a set into itself form a monoid with respect to function composition. More generally, in category theory, the morphisms of an object to itself form a monoid, and, conversely, a monoid may be viewed as a category with a single object.
In computer science and computer programming, the set of strings built from a given set of characters is a free monoid. Transition monoids and syntactic monoids are used in describing finite-state machines. Trace monoids and history monoids provide a foundation for process calculi and concurrent computing.
In theoretical computer science, the study of monoids is fundamental for automata theory (Krohn–Rhodes theory), and formal language theory (star height problem).
See semigroup for the history of the subject, and some other general properties of monoids.
is a free monoid. Transition monoids and syntactic monoids are used in describing finite-state machines. Trace monoids and history monoids provide a foundation...
In abstract algebra, the free monoid on a set is the monoid whose elements are all the finite sequences (or strings) of zero or more elements from that...
In abstract algebra, a monoid ring is a ring constructed from a ring and a monoid, just as a group ring is constructed from a ring and a group. Let R be...
the analogous case of groups) it may be called an abelian semigroup. A monoid is an algebraic structure intermediate between semigroups and groups, and...
synchronization points or thread joins. The trace monoid or free partially commutative monoid is a monoid of traces. In a nutshell, it is constructed as...
computer science, the syntactic monoid M ( L ) {\displaystyle M(L)} of a formal language L {\displaystyle L} is the smallest monoid that recognizes the language...
monoids were first presented by M.W. Shields. History monoids are isomorphic to trace monoids (free partially commutative monoids) and to the monoid of...
topological monoid is a monoid object in the category of topological spaces. In other words, it is a monoid with a topology with respect to which the monoid's binary...
the set of nonnegative integers or the set of integers, but can be any monoid. The direct sum decomposition is usually referred to as gradation or grading...
If it includes the identity function, it is a monoid, called a transformation (or composition) monoid. This is the semigroup analogue of a permutation...
a monoid, one can still use the notion of a generating set S {\displaystyle S} of G {\displaystyle G} . S {\displaystyle S} is a semigroup/monoid generating...
follows that the set of all endomorphisms of X forms a monoid, the full transformation monoid, and denoted End(X) (or EndC(X) to emphasize the category...
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...
important special case is a monoid action or act, in which the semigroup is a monoid and the identity element of the monoid acts as the identity transformation...
In mathematics, the plactic monoid is the monoid of all words in the alphabet of positive integers modulo Knuth equivalence. Its elements can be identified...
algebraic structure is a monoid, usually called the full linear monoid, but occasionally also full linear semigroup, general linear monoid etc. It is actually...
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...
In algebra, a presentation of a monoid (or a presentation of a semigroup) is a description of a monoid (or a semigroup) in terms of a set Σ of generators...
it is in fact a monoid, it is usually referred to as simply a semigroup. It is perhaps most easily understood as the syntactic monoid describing the Dyck...
operation. A monoid homomorphism is a map between monoids that preserves the monoid operation and maps the identity element of the first monoid to that of...
In abstract algebra, an additive monoid ( M , 0 , + ) {\displaystyle (M,0,+)} is said to be zerosumfree, conical, centerless or positive if nonzero elements...
A Cartesian monoid is a monoid, with additional structure of pairing and projection operators. It was first formulated by Dana Scott and Joachim Lambek...
Associated with any semiautomaton is a monoid called the characteristic monoid, input monoid, transition monoid or transition system of the semiautomaton...
multiplicative monoids called the structure sheaf. An affine monoid scheme is a monoidal space that is isomorphic to the spectrum of a monoid, and a monoid scheme...
from linear algebra is the multiplicative monoid of real square matrices of order n (called the full linear monoid). The map which sends a matrix to its transpose...
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...
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