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Generalization of strings in computer science
equivalence under all reorderings. The trace monoid or free partially commutative monoid is a monoid of traces. Traces were introduced by Pierre Cartier and
Trace_monoid
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
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
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
instructions at trace level granularity. The formal mathematical theory of traces is described by trace monoids. The earliest academic publication of trace cache
Trace_cache
Smallest monoid that recognizes a formal language
language with syntactic monoid Z / 2 n Z {\displaystyle \mathbb {Z} /2^{n}\mathbb {Z} } . Trace monoids are examples of syntactic monoids. Marcel-Paul Schützenberger
Syntactic_monoid
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
Topics referred to by the same term
Galinon-Mélénec TRACE, a request method in the HTTP protocol Traces, the equivalence classes of strings of a trace monoid, studied in trace theories of concurrent
Trace
Directed graph representing dependencies
evaluation order 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
Dependency_graph
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
Property of some mathematical operations
commutative if and only if it is a symmetric function of its two arguments) Trace monoid Rice 2011, p. 4. Saracino 2008, p. 11. Hall 1966, pp. 262–263. Lovett
Commutative_property
Replacing subterm in a formula with another term
representation. Trace theory provides a means for discussing multiprocessing in more formal terms, such as via the trace monoid and the history monoid. Rewriting
Rewriting
Formal model in concurrency theory
Oxford University Computing Laboratory." Trace theory, the general theory of traces. Trace monoid and history monoid Ease programming language XC programming
Communicating sequential processes
Communicating_sequential_processes
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
Variant of the notion of the center of a monoid, group, or ring to a category
operation, monoid objects in C {\displaystyle {\mathcal {C}}} are monoidal categories, and the above recovers the Drinfeld center. The categorical trace of a
Center_(category_theory)
monoid on a finite alphabet is compact. A free monoid on a countable alphabet is compact. A finitely generated free group is compact. A trace monoid on
Compact_semigroup
Square matrices satisfy their characteristic equation
polynomial was given by Straubing and a generalization was given using trace monoid theory of Foata and Cartier. The above proofs show that the Cayley–Hamilton
Cayley–Hamilton_theorem
Collection of objects and morphisms
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)
Specific type of Petri net
the different parameters given to the processes/threads. History monoid Trace monoid Johnsonbaugh, Richard; Murata, Tadao (October 1982). "Petri Nets
Marked_graph
for traces can be found in The Book of Traces. A monoid in which Levi's lemma holds is said to have the equidivisibility property. The free monoid of strings
Levi's_lemma
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
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
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
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
Executing several computations during overlapping time periods
Concurrent Object-Oriented Programming (SCOOP) Reo Coordination Language Trace monoids Some of these models of concurrency are primarily intended to support
Concurrent_computing
Family of infinite discrete groups
admits an Artin–Tits presentation. Likewise, an Artin–Tits monoid is a monoid that, as a monoid, admits an Artin–Tits presentation. Alternatively, an Artin–Tits
Artin–Tits_group
In mathematics, invertible homomorphism
3-category Categorified concepts 2-group 2-ring En-ring (Traced)(Symmetric) monoidal category Monoidal functor n-group n-monoid Category Outline Glossary
Isomorphism
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
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
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
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
category theory, a traced monoidal category is a category with some extra structure which gives a reasonable notion of feedback. A traced symmetric monoidal
Traced_monoidal_category
Type of category in mathematics
3-category Categorified concepts 2-group 2-ring En-ring (Traced)(Symmetric) monoidal category Monoidal functor n-group n-monoid Category Outline Glossary
Topos
Generalized object in category theory
Segre embedding. In the category of semi-abelian monoids, the product is given by the history monoid. In the category of Banach spaces and short maps
Product_(category_theory)
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 unit
Simplex_category
Family of approaches for modelling concurrent systems
is then a formal language imposed on a history monoid in a consistent fashion. That is, a history monoid can only record a sequence of events, with synchronization
Process_calculus
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
In mathematics, invariant of square matrices
category of commutative rings to the category of monoids. The first functor maps each ring to the monoid (under matrix multiplication) of the square matrices
Determinant
Aspect of category theory
arrow going between them. The coequalizer of these two functors is the monoid of natural numbers under addition, considered as a one-object category.
Coequalizer
Mathematical category whose hom sets form Abelian groups
same way that a monoid can be viewed as a category with only one object—and forgetting the additive structure of the ring gives us a monoid). In this way
Preadditive_category
Map (arrow) between two objects of a category
3-category Categorified concepts 2-group 2-ring En-ring (Traced)(Symmetric) monoidal category Monoidal functor n-group n-monoid Category Outline Glossary
Morphism
Mathematical category formed by reversing morphisms
completing a semigroup to a monoid, taking the corresponding opposite category, and then possibly removing the unit from that monoid. The category of Boolean
Opposite_category
Generalization of category theory
3-category Categorified concepts 2-group 2-ring En-ring (Traced)(Symmetric) monoidal category Monoidal functor n-group n-monoid Category Outline Glossary
Higher_category_theory
Type of category in category theory
ISBN 0-444-87508-5. "Ct.category theory - is the category commutative monoids cartesian closed?". Backus, John (1981). "Function level programs as mathematical
Cartesian_closed_category
Mathematical category with weak equivalences, fibrations and cofibrations
3-category Categorified concepts 2-group 2-ring En-ring (Traced)(Symmetric) monoidal category Monoidal functor n-group n-monoid Category Outline Glossary
Model_category
Theorem in category theory
3-category Categorified concepts 2-group 2-ring En-ring (Traced)(Symmetric) monoidal category Monoidal functor n-group n-monoid Category Outline Glossary
Lawvere's_fixed-point_theorem
Surjective homomorphism
To see this, suppose that g1 and g2 are two distinct maps from Z to some monoid M. Then for some n in Z, g1(n) ≠ g2(n), so g1(−n) ≠ g2(−n). Either n or
Epimorphism
Type of category in mathematics
3-category Categorified concepts 2-group 2-ring En-ring (Traced)(Symmetric) monoidal category Monoidal functor n-group n-monoid Category Outline Glossary
Elementary_topos
Collection of maps which give the same result
3-category Categorified concepts 2-group 2-ring En-ring (Traced)(Symmetric) monoidal category Monoidal functor n-group n-monoid Category Outline Glossary
Commutative_diagram
Most general completion of a commutative square given two morphisms with same domain
associative algebras for the case of non-commutative rings. In the multiplicative monoid of positive integers Z + {\displaystyle \mathbf {Z} _{+}} , considered as
Pushout_(category_theory)
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
Central object of study in category theory
3-category Categorified concepts 2-group 2-ring En-ring (Traced)(Symmetric) monoidal category Monoidal functor n-group n-monoid Category Outline Glossary
Natural_transformation
Construction in category theory
3-category Categorified concepts 2-group 2-ring En-ring (Traced)(Symmetric) monoidal category Monoidal functor n-group n-monoid Category Outline Glossary
Inverse_limit
Generalization of the kernel of a homomorphism
information for algebraic purposes. Therefore, the notion of kernel studied in monoid theory is slightly different (see #Relationship to algebraic kernels below)
Kernel_(category_theory)
Mathematical set with repetitions allowed
It defines a commutative monoid structure on the finite multisets in a given universe. This monoid is a free commutative monoid, with the universe as a
Multiset
Mathematical concept
3-category Categorified concepts 2-group 2-ring En-ring (Traced)(Symmetric) monoidal category Monoidal functor n-group n-monoid Category Outline Glossary
Limit_(category_theory)
Most general completion of a commutative square given two morphisms with same codomain
as the "intersection" of the two subobjects. Consider the multiplicative monoid of positive integers Z+ as a category with one object. In this category
Pullback_(category_theory)
Type of category in category theory
then a remarkable theorem that the Hom sets naturally admit an abelian monoid structure. A proof of this fact is given below. An additive category may
Additive_category
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
3-category Categorified concepts 2-group 2-ring En-ring (Traced)(Symmetric) monoidal category Monoidal functor n-group n-monoid Category Outline Glossary
Tetracategory
Binary relation in computer science
irreflexive relation ≐ {\displaystyle \doteq } can be defined on the free monoid Σ ∗ {\displaystyle \Sigma ^{*}} of all possible strings of finite length
Dependency_relation
Category-theoretic construction
Y\oplus X.} These properties are formally similar to those of a commutative monoid; a category with finite coproducts is an example of a symmetric monoidal
Coproduct
Category whose objects are R-modules and whose morphisms are module homomorphisms
\otimes } , the category of modules is a symmetric monoidal category. A monoid object of the category of modules over a commutative ring R {\displaystyle
Category_of_modules
Abstract mathematics relationship
3-category Categorified concepts 2-group 2-ring En-ring (Traced)(Symmetric) monoidal category Monoidal functor n-group n-monoid Category Outline Glossary
Equivalence_of_categories
Characterizing property of mathematical constructions
3-category Categorified concepts 2-group 2-ring En-ring (Traced)(Symmetric) monoidal category Monoidal functor n-group n-monoid Category Outline Glossary
Universal_property
Functor type
3-category Categorified concepts 2-group 2-ring En-ring (Traced)(Symmetric) monoidal category Monoidal functor n-group n-monoid Category Outline Glossary
Representable_functor
the free category on Q has only one object, and corresponds to the free monoid on the edges of Q. The category of small categories Cat has a forgetful
Free_category
Special case of colimit in category theory
3-category Categorified concepts 2-group 2-ring En-ring (Traced)(Symmetric) monoidal category Monoidal functor n-group n-monoid Category Outline Glossary
Direct_limit
Category with direct sums and certain types of kernels and cokernels
3-category Categorified concepts 2-group 2-ring En-ring (Traced)(Symmetric) monoidal category Monoidal functor n-group n-monoid Category Outline Glossary
Abelian_category
Concept in mathematics
3-category Categorified concepts 2-group 2-ring En-ring (Traced)(Symmetric) monoidal category Monoidal functor n-group n-monoid Category Outline Glossary
Tensor–hom_adjunction
Injective homomorphism
3-category Categorified concepts 2-group 2-ring En-ring (Traced)(Symmetric) monoidal category Monoidal functor n-group n-monoid Category Outline Glossary
Monomorphism
Category theory
3-category Categorified concepts 2-group 2-ring En-ring (Traced)(Symmetric) monoidal category Monoidal functor n-group n-monoid Category Outline Glossary
Kleisli_category
Category whose hom sets have algebraic structure
the monoidal identity object I of M, being an identity for ⊗ only in the monoid-theoretic sense, and even then only up to canonical isomorphism (λ, ρ).
Enriched_category
Category whose objects are groups and whose morphisms are group homomorphisms
\mathbf {Grp} } the left adjoint functor sending every monoid to the Grothendieck group of that monoid. The forgetful functor U : G r p → S e t {\displaystyle
Category_of_groups
Homological construction in category theory
3-category Categorified concepts 2-group 2-ring En-ring (Traced)(Symmetric) monoidal category Monoidal functor n-group n-monoid Category Outline Glossary
Derived_functor
Set of arguments where two or more functions have the same value
3-category Categorified concepts 2-group 2-ring En-ring (Traced)(Symmetric) monoidal category Monoidal functor n-group n-monoid Category Outline Glossary
Equaliser_(mathematics)
3-category Categorified concepts 2-group 2-ring En-ring (Traced)(Symmetric) monoidal category Monoidal functor n-group n-monoid Category Outline Glossary
Essentially surjective functor
Essentially_surjective_functor
Type of quotient object in mathematics
Monoids and groups may be regarded as categories with one object. In this case the quotient category coincides with the notion of a quotient monoid or
Quotient_category
Quotient space of a codomain of a linear map by the map's image
3-category Categorified concepts 2-group 2-ring En-ring (Traced)(Symmetric) monoidal category Monoidal functor n-group n-monoid Category Outline Glossary
Cokernel
Functors which are surjective and injective on hom-sets
3-category Categorified concepts 2-group 2-ring En-ring (Traced)(Symmetric) monoidal category Monoidal functor n-group n-monoid Category Outline Glossary
Full_and_faithful_functors
Aspect of category theory in mathematics
3-category Categorified concepts 2-group 2-ring En-ring (Traced)(Symmetric) monoidal category Monoidal functor n-group n-monoid Category Outline Glossary
Rig_category
3-category Categorified concepts 2-group 2-ring En-ring (Traced)(Symmetric) monoidal category Monoidal functor n-group n-monoid Category Outline Glossary
Conservative_functor
Functor that preserves short exact sequences
3-category Categorified concepts 2-group 2-ring En-ring (Traced)(Symmetric) monoidal category Monoidal functor n-group n-monoid Category Outline Glossary
Exact_functor
Category theory constructs
3-category Categorified concepts 2-group 2-ring En-ring (Traced)(Symmetric) monoidal category Monoidal functor n-group n-monoid Category Outline Glossary
Kan_extension
Category whose objects are rings and whose morphisms are ring homomorphisms
over Ab (the category of abelian groups) or over Mon (the category of monoids). Specifically, there are forgetful functors A : Ring → Ab M : Ring → Mon
Category_of_rings
Graphical representation of a morphism
diagrams. Let the Kleene star X ⋆ {\displaystyle X^{\star }} denote the free monoid, i.e. the set of lists with elements in a set X {\displaystyle X} . A monoidal
String_diagram
Higher categorical generalization of a topos
3-category Categorified concepts 2-group 2-ring En-ring (Traced)(Symmetric) monoidal category Monoidal functor n-group n-monoid Category Outline Glossary
∞-topos
Category whose objects are sets and whose morphisms are functions
3-category Categorified concepts 2-group 2-ring En-ring (Traced)(Symmetric) monoidal category Monoidal functor n-group n-monoid Category Outline Glossary
Category_of_sets
Generalization of a category
3-category Categorified concepts 2-group 2-ring En-ring (Traced)(Symmetric) monoidal category Monoidal functor n-group n-monoid Category Outline Glossary
Quasi-category
Category enriched over the category of simplicial sets
3-category Categorified concepts 2-group 2-ring En-ring (Traced)(Symmetric) monoidal category Monoidal functor n-group n-monoid Category Outline Glossary
Simplicially enriched category
Simplicially_enriched_category
Correspondence between properties of a category and its opposite
3-category Categorified concepts 2-group 2-ring En-ring (Traced)(Symmetric) monoidal category Monoidal functor n-group n-monoid Category Outline Glossary
Dual_(category_theory)
Mathematical construction used in homotopy theory
3-category Categorified concepts 2-group 2-ring En-ring (Traced)(Symmetric) monoidal category Monoidal functor n-group n-monoid Category Outline Glossary
Simplicial_set
Construction in category theory
3-category Categorified concepts 2-group 2-ring En-ring (Traced)(Symmetric) monoidal category Monoidal functor n-group n-monoid Category Outline Glossary
Cone_(category_theory)
3-category Categorified concepts 2-group 2-ring En-ring (Traced)(Symmetric) monoidal category Monoidal functor n-group n-monoid Category Outline Glossary
Stable_∞-category
conditions like associativity. For example, a monoid object in Set is a usual monoid (unital semigroup) and a monoid object in R-mod is an associative algebra
Glossary_of_category_theory
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
3-category Categorified concepts 2-group 2-ring En-ring (Traced)(Symmetric) monoidal category Monoidal functor n-group n-monoid Category Outline Glossary
Lift_(mathematics)
3-category Categorified concepts 2-group 2-ring En-ring (Traced)(Symmetric) monoidal category Monoidal functor n-group n-monoid Category Outline Glossary
3-category
Hypothesis in mathematical category theory
3-category Categorified concepts 2-group 2-ring En-ring (Traced)(Symmetric) monoidal category Monoidal functor n-group n-monoid Category Outline Glossary
Homotopy_hypothesis
Mathematics construct
3-category Categorified concepts 2-group 2-ring En-ring (Traced)(Symmetric) monoidal category Monoidal functor n-group n-monoid Category Outline Glossary
Comma_category
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TRACE MONOID
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TRACE MONOID
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