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Algebraic structure with an associative operation and an identity element
is a free monoid. Transition monoids and syntactic monoids are used in describing finite-state machines. Trace monoids and history monoids provide a foundation
Monoid
Topics referred to by the same term
Look up monoid in Wiktionary, the free dictionary. A monoid is an algebraic structure. Monoid may also refer to: Monoid (category theory), a mathematical
Monoid_(disambiguation)
Concept in mathematics
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
Free_monoid
Concept in mathematics
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
Topological_monoid
Algebraic structure
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
Monoid_ring
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
Presentation_of_a_monoid
Type of algebraic structure
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
Graded_ring
Mathematical concept in category theory
In category theory, a branch of mathematics, a monoid (or monoid object, or internal monoid, or algebra) ( M , μ , η ) {\displaystyle (M,\mu ,\eta )} in
Monoid_(category_theory)
Smallest monoid that recognizes a formal language
computer science, the syntactic monoid M ( L ) {\displaystyle M(L)} of a formal language L {\displaystyle L} is the minimal monoid that recognizes the language
Syntactic_monoid
A Cartesian monoid is a monoid, with additional structure of pairing and projection operators. It was first formulated by Dana Scott and Joachim Lambek
Cartesian_monoid
Algebraic structure
not a monoid. Positive integers with addition form a commutative semigroup that is not a monoid, whereas the non-negative integers do form a monoid. A semigroup
Semigroup
Generalization of strings in computer science
complete equivalence under all reorderings. The trace monoid or free partially commutative monoid is a monoid of traces. Traces were introduced by Pierre Cartier
Trace_monoid
Concept in abstract algebra
In abstract algebra, an additive monoid ( M , 0 , + ) {\displaystyle (M,0,+)} is said to be zerosumfree, conical, centerless or positive if nonzero elements
Zerosumfree_monoid
monoids were first presented by M.W. Shields. History monoids are isomorphic to trace monoids (free partially commutative monoids) and to the monoid of
History_monoid
Concept in abstract algebra
In mathematics, a refinement monoid is a commutative monoid M such that for any elements a0, a1, b0, b1 of M such that a0+a1=b0+b1, there are elements
Refinement_monoid
Truncating subtraction on natural numbers, or a generalization thereof
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
Monus
In mathematics, the Chinese monoid is a monoid generated by a totally ordered alphabet with the relations cba = cab = bca for every a ≤ b ≤ c. An algorithm
Chinese_monoid
Theoretical object in mathematics
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
Field_with_one_element
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
Rational_monoid
If it includes the identity function, it is a monoid, called a transformation (or composition) monoid. This is the semigroup analogue of a permutation
Transformation_semigroup
Action of a semigroup on a set
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
Semigroup_action
Category admitting tensor products
category may also be viewed as a "categorification" of an underlying monoid, namely the monoid whose elements are the isomorphism classes of the category's objects
Monoidal_category
a factorisation of a free monoid is a sequence of subsets of words with the property that every word in the free monoid can be written as a concatenation
Monoid_factorisation
Monoid of all words in the alphabet of positive integers modulo Knuth equivalence
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
Plactic_monoid
Mathematical object that generalizes the standard notions of sets and functions
Any monoid can be understood as a special sort of category (with a single object whose self-morphisms are represented by the elements of the monoid), and
Category_(mathematics)
Finitelt generated commutative 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
Affine_monoid
Semigroup in abstract algebra
has in the general linear group (which is a subgroup of the full linear monoid). However, for an arbitrary matrix, AAT does not equal the identity element
Semigroup_with_involution
Associated with any semiautomaton is a monoid called the characteristic monoid, input monoid, transition monoid or transition system of the semiautomaton
Semiautomaton
Abstract algebra concept
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
Generating_set_of_a_group
Group of 𝑛 × 𝑛 invertible matrices
algebraic structure is a monoid, usually called the full linear monoid, but occasionally also full linear semigroup, general linear monoid etc. It is actually
General_linear_group
Self-self morphism
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
Endomorphism
1966 Doctor Who serial
to discover the humans have become subservient to their slave race, the Monoids. Producer John Wiles conceived of the spaceship, and story editor Donald
The_Ark_(Doctor_Who)
Abelian group extending a commutative monoid
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
Grothendieck_group
String rewriting system
coincides with the presentation of a monoid. Thus they constitute a natural framework for solving the word problem for monoids and groups. An SRS can be defined
Semi-Thue_system
examples of Frobenioids are essentially monoids. If M is a commutative monoid, it is acted on naturally by the monoid N of positive integers under multiplication
Frobenioid
Element of algebraic structure
element of an algebraic structure such as a monoid that has several desirable properties. Formally, if M is a monoid, then an element Δ of M is said to be a
Garside_element
Orientation-preserving mapping class group of the torus
group is the dyadic monoid, which is the monoid of all strings of the form STn1STn2STn3... for positive integers ni. This monoid occurs naturally in the
Modular_group
Algebraic ring that need not have additive negative elements
arises as the function composition of endomorphisms over any commutative monoid. Some authors define semirings without the requirement for there to be a
Semiring
Continuous fractal curve obtained as the image of Cantor space
are given by the monoid that describes the symmetries of the infinite binary tree or Cantor space. This so-called period-doubling monoid is a subset of
De_Rham_curve
Design pattern in functional programming to build generic types
to the category of monoids. Here the task for the programmer is to construct an appropriate monoid, or perhaps to choose a monoid from a library. The
Monad (functional programming)
Monad_(functional_programming)
Generalised alphabetical order
separate sorting algorithm. The monoid of words over an alphabet A is the free monoid over A. That is, the elements of the monoid are the finite sequences (words)
Lexicographic_order
Finite, ordered collection of items
Lists form a monoid under the append operation. The identity element of the monoid is the empty list, nil. In fact, this is the free monoid over the set
List_(abstract_data_type)
Variant of the notion of the center of a monoid, group, or ring to a category
mathematician Vladimir Drinfeld) is a variant of the notion of the center of a monoid, group, or ring to a category. The center of a monoidal category C = ( C
Center_(category_theory)
category theory, a (strict) n-monoid is an n-category with only one 0-cell. In particular, a 1-monoid is a monoid and a 2-monoid is a strict monoidal category
N-monoid
Theory of trace monoids
definition of the free partially commutative monoid or trace monoid, or equivalently, the history monoid, which provides a concrete algebraic foundation
Trace_theory
Directed graph representing dependencies
as well. An acyclic dependency graph corresponds to a trace of a trace monoid as follows: A function ϕ : S → Σ {\displaystyle \phi :S\to \Sigma } labels
Dependency_graph
Property of operations
{\displaystyle x\cdot x=x} for all x ∈ S {\displaystyle x\in S} . In the monoid ( N , × ) {\displaystyle (\mathbb {N} ,\times )} of the natural numbers
Idempotence
Finite-state machine
Repeated function composition forms a monoid. For the transition functions, this monoid is known as the transition monoid, or sometimes the transformation
Deterministic finite automaton
Deterministic_finite_automaton
Unary operation on string sets
elements belong to V; in mathematics, it is more commonly known as the free monoid construction. The Kleene star operator on a language L generates another
Kleene_star
Ring that is also a vector space or a module
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 abelian
Associative_algebra
General theory of mathematical structures
the case. For example, a monoid may be viewed as a category with a single object, whose morphisms are the elements of the monoid. The second fundamental
Category_theory
Function that returns its argument unchanged
the monoid of all functions from X {\displaystyle X} to X {\displaystyle X} (under function composition). Since the identity element of a monoid is unique
Identity_function
Special kind of semigroup in mathematics
not in the set. Numerical semigroups are commutative monoids and are also known as numerical monoids. The definition of numerical semigroup is intimately
Numerical_semigroup
Structure-preserving map between two algebraic structures of the same type
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
Homomorphism
Operation in algebra and mathematics
considered at least in two ways: A monad as a generalized monoid; this is clear since a monad is a monoid in a certain category, A monad as a tool for studying
Monad_(category_theory)
Type of semigroup
positive integer n such that xn = xn+1. An aperiodic monoid is an aperiodic semigroup which is a monoid. A finite semigroup is aperiodic if and only if it
Aperiodic_semigroup
Operation on mathematical functions
structure of a monoid, called a transformation monoid or (much more seldom) a composition monoid. In general, transformation monoids can have remarkably
Function_composition
in automata theory, a rational set of a monoid is an element of the minimal class of subsets of this monoid that contains all finite subsets and is closed
Rational_set
Sequence of characters, data type
operation form a monoid, the free monoid generated by Σ {\displaystyle \Sigma } . In addition, the length function defines a monoid homomorphism from
String_(computer_science)
Set with operations obeying given axioms
(juxtaposition) as is done for ordinary multiplication of real numbers. Group: a monoid with a unary operation (inverse), giving rise to inverse elements. Abelian
Algebraic_structure
Function with unusual fractal properties
These two operators may be repeatedly combined, forming a monoid. A general element of the monoid is then S a 1 R S a 2 R S a 3 ⋯ {\displaystyle
Minkowski's question-mark function
Minkowski's_question-mark_function
Mathematical theorem
the structures are the same, and the resulting magma is a commutative monoid. This can then be used to prove the commutativity of the higher homotopy
Eckmann–Hilton_argument
In mathematics, an algebraic structure
is an algebraic structure that is simultaneously a lattice x ≤ y and a monoid x•y that admits operations x\z and z/y, loosely analogous to division or
Residuated_lattice
Whole of an object being mathematically similar to part of itself
algebraic structure of a monoid. When the set S has only two elements, the monoid is known as the dyadic monoid. The dyadic monoid can be visualized as an
Self-similarity
Algebra where division is always defined
multiplication are not a group but respectively a commutative monoid and a commutative monoid with involution. A wheel is an algebraic structure ( W , 0
Wheel_theory
Number used for counting
(\mathbb {N} ,+)} is a commutative monoid with identity element 0. It is a free monoid on one generator. This commutative monoid satisfies the cancellation property
Natural_number
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
Bicyclic_semigroup
Generalization of additive and multiplicative inverses
order. A monoid is a set with an associative operation that has an identity element. The invertible elements in a monoid form a group under monoid operation
Inverse_element
Algebraic structure
and x•z ≤ y•z for all x, y, z in S. An ordered monoid and an ordered group are, respectively, a monoid or a group that are endowed with a partial order
Ordered_semigroup
Algebraic structure
variables with coefficients in the ring R is the monoid ring R[N], where the monoid N is the free monoid on n letters, also known as the set of all strings
Polynomial_ring
Group with a compatible partial order
In abstract algebra, a partially ordered group is a group (G, +) equipped with a partial order "≤" that is translation-invariant; in other words, "≤" has
Partially_ordered_group
Group that has only one element
\mathrm {e} \cdot \mathrm {e} =\mathrm {e} } . The similarly defined trivial monoid is also a group since its only element is its own inverse, and is hence
Trivial_group
Surjective homomorphism
epimorphisms fail to be surjective. A few examples are: In the category of monoids, Mon, the inclusion map N → Z is a non-surjective epimorphism. To see this
Epimorphism
Continuous function that is not absolutely continuous
monoid M is then the monoid of all such finite-length left-right moves. Writing γ ∈ M {\displaystyle \gamma \in M} as a general element of the monoid
Cantor_function
Function with a multiplicative scaling behaviour
numbers can be replaced by the more general notion of a monoid. Let M {\displaystyle M} be a monoid with identity element 1 ∈ M , {\displaystyle 1\in M,}
Homogeneous_function
Algebraic structure with addition and multiplication
such that a + (−a) = 0 (that is, −a is the additive inverse of a). R is a monoid under multiplication, meaning that: (a · b) · c = a · (b · c) for all a
Ring_(mathematics)
Two-dimensional manifold
connected sums, the closed surfaces up to homeomorphism form a commutative monoid under the operation of connected sum, as indeed do manifolds of any fixed
Surface_(topology)
Set with associative invertible operation
structure is called a monoid. The natural numbers N {\displaystyle \mathbb {N} } (including zero) under addition form a monoid, as do the nonzero integers
Group_(mathematics)
Equivalence relation in algebra
cannot be done with, for example, monoids, so the study of congruence relations plays a more central role in monoid theory. The general notion of a congruence
Congruence_relation
Branch of mathematics
abelian monoid into an abelian group is a necessary ingredient for defining K-theory since all definitions start by constructing an abelian monoid from a
K-theory
Mapping between categories
object is the same thing as a monoid: the morphisms of a one-object category can be thought of as elements of the monoid, and composition in the category
Functor
Problem in topology
operations comprise an operator monoid called the Kuratowski monoid where the monoid product is function composition. This monoid, which can be used to classify
Kuratowski's closure-complement problem
Kuratowski's_closure-complement_problem
finite (ordered) monoids is a variety of finite (ordered) semigroups whose elements are monoids. That is, it is a class of (ordered) monoids satisfying the
Variety_of_finite_semigroups
Relationship between two functors abstracting many common constructions
a right adjoint to F. From monoids and groups to rings. The integral monoid ring construction gives a functor from monoids to rings. This functor is left
Adjoint_functors
Algebra with unique prime factorization
endowed with the above product is a commutative semigroup and in fact a monoid: the identity element is the fractional ideal R. For any fractional ideal
Dedekind_domain
Branch of mathematics that studies algebraic structures
Transformation semigroup Monoid Aperiodic monoid Free monoid Monoid (category theory) Monoid factorisation Syntactic monoid Group (mathematics) Lagrange's
List of abstract algebra topics
List_of_abstract_algebra_topics
Algebraic structure with a binary operation
commutativity Commutative magma: A magma with commutativity. Commutative monoid: A monoid with commutativity. Abelian group: A group with commutativity. A magma
Magma_(algebra)
Construction providing a total order on a free monoid
provide a unique monoid factorisation of the free monoid. They are also totally ordered, and thus provide a total order on the monoid. This is analogous
Hall_word
Special type of element of a set
additive notation zero may, quite naturally, denote the neutral element of a monoid. In this article "zero element" and "absorbing element" are synonymous.
Absorbing_element
Special objects used in (mathematical) category theory
notion of final object (respectively, initial object). The endomorphism monoid of an initial or terminal object I is trivial: End(I) = Hom(I, I) = { idI
Initial_and_terminal_objects
Concept in category theory
commutative diagrams: If ( M , μ , ϵ ) {\displaystyle (M,\mu ,\epsilon )} is a monoid object in C {\displaystyle C} , then ( F M , F μ ∘ ϕ M , M , F ϵ ∘ ϕ ) {\displaystyle
Monoidal_functor
Semi-decision algorithm for transforming a set of equations
resultant rewriting system to be confluent. Consider a finitely presented monoid M = ⟨ X ∣ R ⟩ {\displaystyle M=\langle X\mid R\rangle } where X is a finite
Knuth–Bendix completion algorithm
Knuth–Bendix_completion_algorithm
Type of classification in algebra
abelian. Archimedean groups can be generalised to Archimedean monoids, linearly ordered monoids that obey the Archimedean property. Examples include the natural
Archimedean_group
Property of some mathematical operations
semigroup is a semigroup whose operation is commutative; a commutative monoid is a monoid whose operation is commutative; a commutative group or abelian group
Commutative_property
Class of algebraic structures
groups form a variety of algebras, as do the abelian groups, the rings, the monoids etc. According to Birkhoff's theorem, a class of algebraic structures of
Variety_(universal_algebra)
Commutative ring with no zero divisors other than zero
which the set of nonzero elements is a commutative monoid under multiplication (because a monoid must be closed under multiplication). An integral domain
Integral_domain
together with the empty word ϵ {\displaystyle \epsilon } defines a free monoid, the monoid of the words on I {\displaystyle I} , which is one of the simplest
Basis_(universal_algebra)
Embedding of categories into functor categories
3-category Categorified concepts 2-group 2-ring En-ring (Traced)(Symmetric) monoidal category Monoidal functor n-group n-monoid Category Outline Glossary
Yoneda_lemma
Generalization of category
the monoid M = ({T, F}, ∧, T). As a category this is presented with two objects {T, F} and single morphism g: F → T. We can reinterpret this monoid as
2-category
exactly a module over a monoid in C a t {\displaystyle {\mathsf {Cat}}} . For example, S {\displaystyle S} acts on itself via the monoid operation ⊗ {\displaystyle
Monoidal_category_action
MONOID
MONOID
MONOID
MONOID
Girl/Female
Biblical
Who humbles thee, who answers thee.
Boy/Male
Hindu
Compassionate
Girl/Female
Tamil
Lord Krishna
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Telugu
Self Determination Ooth; Taking a Plage; Aim; Promise to do Something; Motive; Will
Boy/Male
Hindu, Indian, Swedish
God is Merciful
Girl/Female
Hebrew
Life.
Boy/Male
Tamil
Tilakarathna | தீலாகாரதநா
Nama
Female
Swedish
 Old Swedish form of Greek Aikaterine, KATERIN means "pure." Compare with another form of Katerin.
Girl/Female
Muslim/Islamic
Opinions
Boy/Male
Arabic
Reality; Sincerity
MONOID
MONOID
MONOID
MONOID
MONOID