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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
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
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
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
Property of some mathematical operations
statistics (for commutativity in physics) Quasi-commutative property Trace monoid Rice 2011, p. 4. Saracino 2008, p. 11. Hall 1966, pp. 262–263. Lovett
Commutative_property
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
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
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
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
Ability to execute a task in a non-serial manner
Concurrent Object-Oriented Programming (SCOOP) Reo Coordination Language Trace monoids Some of these models of concurrency are primarily intended to support
Concurrency (computer science)
Concurrency_(computer_science)
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)
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
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)
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
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
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
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
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
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
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
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
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
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
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
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
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
Mathematical category
3-category Categorified concepts 2-group 2-ring En-ring (Traced)(Symmetric) monoidal category Monoidal functor n-group n-monoid Category Outline Glossary
Topos
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
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
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
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
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
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
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
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
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)
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
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 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)
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)
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
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
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
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
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)
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
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
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
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
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
3-category Categorified concepts 2-group 2-ring En-ring (Traced)(Symmetric) monoidal category Monoidal functor n-group n-monoid Category Outline Glossary
Tetracategory
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)
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
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
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
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
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
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
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
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
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
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
Relation of categories 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
Isomorphism_of_categories
3-category Categorified concepts 2-group 2-ring En-ring (Traced)(Symmetric) monoidal category Monoidal functor n-group n-monoid Category Outline Glossary
Localization_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
Conservative_functor
Category whose objects and morphisms are inside a bigger category
3-category Categorified concepts 2-group 2-ring En-ring (Traced)(Symmetric) monoidal category Monoidal functor n-group n-monoid Category Outline Glossary
Subcategory
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
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
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
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
Applications 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
Applied_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
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
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
In mathematics, collection of classes
3-category Categorified concepts 2-group 2-ring En-ring (Traced)(Symmetric) monoidal category Monoidal functor n-group n-monoid Category Outline Glossary
Conglomerate_(mathematics)
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
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)
Branch of mathematics that studies abstract algebraic structures
generalization is to monoids, which are categories with one object. Groups are monoids for which every morphism is invertible. General monoids have representations
Representation_theory
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
Essentially surjective functor
Essentially_surjective_functor
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
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
Concept 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
Fibred_category
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
End_(category_theory)
Concept 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
Forgetful_functor
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
Algebraic structure with "nice" duality properties
:A\to I} such that ( A , μ , η ) {\displaystyle (A,\mu ,\eta )\,} is a monoid object in C, ( A , δ , ε ) {\displaystyle (A,\delta ,\varepsilon )} is a
Frobenius_algebra
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
Connects set theory with 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
Categorification
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
3-category Categorified concepts 2-group 2-ring En-ring (Traced)(Symmetric) monoidal category Monoidal functor n-group n-monoid Category Outline Glossary
N-group_(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
2-group
TRACE MONOID
TRACE MONOID
Surname or Lastname
English (Kent)
English (Kent) : perhaps a variant of Treece.Altered spelling of German Treis, a topographic name for someone who lived by or owned an uncultivated piece of land used as pasture, from Middle Low German drīsch ‘fallow land’, or a habitational name from a place named with this word (in Hessian dialect treis), in Hesse or on the Mosel river. Alternatively, in some instances it may be from a short form of the personal name Andreas (see Andrew).
Boy/Male
Anglo Saxon American English French
Brave.
Boy/Male
Anglo Saxon American Greek
Brave.
Girl/Female
Latin American English Irish
Grace.
Female
English
Feminine variant spelling of English unisex Tracy, TRACEE means "place of Thracius."
Boy/Male
American, Anglo, Australian, British, Chinese, English, French
Fighter; Brave
Female
English
Feminine variant spelling of English unisex Tracy, TRACI means "place of Thracius."
Male
English
Variant spelling of English unisex Tracy, TRACEY means "place of Thracius."
Male
English
Short form of English unisex Tracy, TRACE means "place of Thracius."
Girl/Female
English American
from Thracia.
Boy/Male
Anglo Saxon American Latin Greek English French
Brave.
Girl/Female
Greek American French
Reaper; from Therasia.
Girl/Female
Greek American
Reap; from Therasia.
Surname or Lastname
English
English : perhaps a variant of Treece.
Male
English
English surname transferred to unisex forename use, from a Norman baronial name TRACY means "place of Thracius."
Female
English
Feminine variant spelling of English unisex Tracy, TRACIE means "place of Thracius."
Girl/Female
English
from Thracia.
Surname or Lastname
English
English : probably from Middle English, Old French brace ‘arm’, also denoting a piece of armor covering the arm. In most cases it is probably a metonymic occupational name for a maker or seller of armor, specifically armor designed to protect the upper arms, but it could also have been a nickname for someone with strong arms (compare Armstrong) or a deformed or otherwise noticeable arm.
Girl/Female
American, Arabic, Australian, British, Chinese, Christian, Danish, English, French, German, Gujarati, Indian, Irish, Jamaican, Latin, Muslim, Portuguese, Swedish
Mercy; God's Favor; Grace; Grace of God; Kindness; Thanks; Love; Favour; Blessing; Charm; Good will
Surname or Lastname
English
English : nickname from Middle English, Old French grace ‘charm’, ‘pleasantness’ (Latin gratia).English : from the female personal name Grace, which was popular in the Middle Ages. This seems in the first instance to have been from a Germanic element grīs ‘gray’ (see Grice 1), but was soon associated by folk etymology with the Latin word meaning ‘charm’.
TRACE MONOID
TRACE MONOID
Girl/Female
Hindu, Indian, Marathi, Traditional
Proud of Herself
Boy/Male
Indian, Sanskrit
Lord of the Earth; King
Boy/Male
Muslim
Pathan. Leader.
Girl/Female
Arabic
The Woman
Male
Portuguese
Portuguese form of Latin Reynaldus, RONALDO means "wise ruler."
Girl/Female
Indian, Sanskrit
Embellishment; Precious
Boy/Male
Muslim
Righteousness
Boy/Male
Australian, Polish
Famous Warrior; Fame and War
Boy/Male
Arabic, Muslim, Sindhi
One who is Led; Obedient; Conducted
Boy/Male
Russian
Earth-lover.
TRACE MONOID
TRACE MONOID
TRACE MONOID
TRACE MONOID
TRACE MONOID
v.
Continuity or extension of anything; as, the tract of speech.
imp. & p. p.
of Trace
v. t.
To run a race with.
n.
Course; way; as, the track of a comet.
n.
A mark left by something that has passed along; as, the track, or wake, of a ship; the track of a meteor; the track of a sled or a wheel.
v. t.
To follow the tracks or traces of; to pursue by following the marks of the feet; to trace; to trail; as, to track a deer in the snow.
v. t.
To cause to contend in a race; to drive at high speed; as, to race horses.
n.
One who, or that which, traces.
n.
A tract or area, as of land.
v.
The trade winds.
v. t.
A mark left by anything passing; a track; a path; a course; a footprint; a vestige; as, the trace of a carriage or sled; the trace of a deer; a sinuous trace.
v. t.
To trace out; to track; also, to draw out; to protact.
v.
Track; trace.
v.
A company of men engaged in the same occupation; thus, booksellers and publishers speak of the customs of the trade, and are collectively designated as the trade.
n.
A mark or impression left by the foot, either of man or beast; trace; vestige; footprint.
v. t.
To supply with heavenly grace.
v. t.
To add grace notes, cadenzas, etc., to.
v. t.
Hence, to follow the trace or track of.
v. t.
To trace by scent; to track; -- a hunting term.
v. t.
To mark out; to draw or delineate with marks; especially, to copy, as a drawing or engraving, by following the lines and marking them on a sheet superimposed, through which they appear; as, to trace a figure or an outline; a traced drawing.