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Copy of a directed graph with redundant edges removed
In the mathematical field of graph theory, a transitive reduction of a directed graph D is another directed graph with the same vertices and as few edges
Transitive_reduction
Directed graph with no directed cycles
contains a longer directed path from u to v. Like the transitive closure, the transitive reduction is uniquely defined for DAGs. In contrast, for a directed
Directed_acyclic_graph
Smallest transitive relation containing a given binary relation
p. 337). We have R+ = R if, and only if, R itself is transitive. Conversely, transitive reduction reduces a minimal relation S from a given relation R
Transitive_closure
Type of binary relation
In mathematics, a binary relation R on a set X is transitive if, for all elements a, b, c in X, whenever R relates a to b and b to c, then R also relates
Transitive_relation
Visual depiction of a partially ordered set
a finite partially ordered set, in the form of a drawing of its transitive reduction. Concretely, for a partially ordered set ( S , ≤ ) {\displaystyle
Hasse_diagram
Method for partitioning partial orders into levels
directed consistently downwards. For a partial ordering given by its transitive reduction (covering relation), the Coffman–Graham algorithm can be implemented
Coffman–Graham_algorithm
Transformations induced by a mathematical group
{\displaystyle g\cdot x=y} . The action is simply transitive (or sharply transitive, or regular) if it is both transitive and free. This means that given x , y ∈
Group_action
Node ordering for directed acyclic graphs
for which x ≤ y. An alternative way of doing this is to use the transitive reduction of the partial ordering; in general, this produces DAGs with fewer
Topological_sorting
Five sporadic simple groups
M23 and M24 introduced by Émile Mathieu (1861, 1873). They are multiply transitive permutation groups on 11, 12, 22, 23 or 24 objects. They are the first
Mathieu_group
A transitive reduction of a graph is a minimal graph having the same transitive closure; directed acyclic graphs have a unique transitive reduction. A
Glossary_of_graph_theory
Mathematical set with an ordering
node and each element of < {\displaystyle <} to be an edge. The transitive reduction of this DAG is then the Hasse diagram. Similarly this process can
Partially_ordered_set
Directed graph representing dependencies
{\displaystyle G=(S,T)} with T ⊆ R {\displaystyle T\subseteq R} the transitive reduction of R. For example, assume a simple calculator. This calculator supports
Dependency_graph
Visual representation of a mathematical relationship
partially ordered set, forming a drawing of the partial order's transitive reduction. Concretely, one represents each element of the set as a vertex on
Mathematical_diagram
Mathematical relation inside orderings
a partially ordered set is finite, its covering relation is the transitive reduction of the partial order relation. Such partially ordered sets are therefore
Covering_relation
Type of graph in mathematics
partial order. Conversely, in a diamond-free partial order, the transitive reduction identifies a directed acyclic graph in which the subgraph reachable
Multitree
Whether one vertex can be reached from another in a graph
defined in this way, for instance as the reachability relation of its transitive reduction. A noteworthy consequence of this is that since partial orders are
Reachability
Pattern relating to the subject and object of verbs
intransitive verb behaves like the object of a transitive verb, and differently from the agent of a transitive verb. In ergative–absolutive languages with
Ergative–absolutive_alignment
Mathematical-logic system
the number of β-reduction steps taken by normal order reduction to reduce a term is a reasonable time cost model, that is, the reduction can be simulated
Lambda_calculus
Formal system for transcribing expressions into equivalent terms
}}} is the transitive closure of → {\displaystyle \rightarrow } . → ∗ {\displaystyle {\stackrel {*}{\rightarrow }}} is the reflexive transitive closure of
Abstract_rewriting_system
subset of a reduction ordering. Conversely, for every terminating term rewriting system, the transitive closure of (::=) is a reduction ordering, which
Rewrite_order
Directed graph describing citations in documents
James R Clough; Jamie Gollings; Tamar V Loach; Tim S Evans (2015). "Transitive reduction of citation networks". Journal of Complex Networks. 3 (2): 189–203
Citation_graph
Smallest complete lattice containing a partial order
to this version of the Dedekind–MacNeille completion problem. The transitive reduction or covering graph of the Dedekind–MacNeille completion describes
Dedekind–MacNeille_completion
Glossary of terms used in branch of mathematics
a finite partially ordered set, in the form of a drawing of its transitive reduction. Homogeneous relation. A homogeneous relation on a set X {\displaystyle
Glossary_of_order_theory
Transformation of one computational problem to another
complexity theory, a reduction is an algorithm for transforming one problem into another problem. A sufficiently efficient reduction from one problem to
Reduction_(complexity)
Type of computational algorithm
{\displaystyle g\circ f} . This allows the concept of logspace reduction to be transitive. Given two logspace transducers, their composition is still a
Log-space_reduction
Planar directed acyclic graph
forms a two-dimensional complete lattice, whose Hasse diagram is the transitive reduction of the given graph. Conversely, the Hasse diagram of every two-dimensional
St-planar_graph
Binary relation that relates every element to itself
property or is said to possess reflexivity. Along with symmetry and transitivity, reflexivity is one of three properties defining equivalence relations
Reflexive_relation
Category of formal programming language semantics
symmetric-transitive-reflexive closure—is a sound reasoning principle for these languages. However, in practice, most applications of reduction semantics
Operational_semantics
Concept in computability theory
In computability theory, a Turing reduction from a decision problem A {\displaystyle A} to a decision problem B {\displaystyle B} is an oracle machine
Turing_reduction
Type of Turing reduction
and computational complexity theory, a many-one reduction (also called mapping reduction) is a reduction that converts instances of one decision problem
Many-one_reduction
Type of group in abstract algebra
inclusion map S5 → S6 as a transitive subgroup; the obvious inclusion map Sn → Sn+1 fixes a point and thus is not transitive. This yields the outer automorphism
Symmetric_group
Type of geometry
program. More specifically, it is a homogeneous space X together with a transitive action on X by a Lie group G, which acts as the symmetry group of the
Klein_geometry
Replacing subterm in a formula with another term
ARS. → ∗ {\displaystyle {\overset {*}{\rightarrow }}} is the reflexive transitive closure of → {\displaystyle \rightarrow } . ↔ {\displaystyle \leftrightarrow
Rewriting
Mapping a graph onto itself without changing edge-vertex connectivity
graph that is edge-transitive but not vertex-transitive. A half-transitive graph is a graph that is vertex-transitive and edge-transitive but not symmetric
Graph_automorphism
Reflexive and transitive binary relation
a preorder or quasiorder is a binary relation that is reflexive and transitive. The name preorder is meant to suggest that preorders are almost partial
Preorder
this way have been called vertex series parallel graphs, and their transitive reductions (the graphs of the covering relations of the partial order) are
Series-parallel_partial_order
Any individual whose preferences satisfy four axioms has a utility function
reflexivity.[clarification needed] Transitivity assumes that preferences are consistent across any three options: Axiom 2 (Transitivity) If L ⪰ M {\displaystyle
Von Neumann–Morgenstern utility theorem
Von_Neumann–Morgenstern_utility_theorem
Method of comparing problems by transforming one into another in computability theory
natural numbers that is Reflexive: Every set is reducible to itself. Transitive: If a set A {\displaystyle A} is reducible to a set B {\displaystyle B}
Reduction (computability theory)
Reduction_(computability_theory)
Concept in mathematics
In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group G over a
Reductive_group
Theorem in theoretical computer science
two lambda expressions. If β-reduction is denoted by → β {\displaystyle \rightarrow _{\beta }} and its reflexive, transitive closure by ↠ β {\displaystyle
Church–Rosser_theorem
Concept of sentence structure in linguistics
intransitive verbs are treated like subjects of transitive verbs, and are distinguished from objects of transitive verbs in basic clause constructions. Nominative–accusative
Nominative–accusative alignment
Nominative–accusative_alignment
Number and type of arguments controlled by a linguistic predicate
Valency is related, though not identical, to subcategorization and transitivity, which count only object arguments – valency counts all arguments, including
Valency_(linguistics)
Sporadic simple group
sporadic groups and was introduced by Mathieu (1861, 1873). It is a 3-fold transitive permutation group on 22 objects. The Schur multiplier of M22 is cyclic
Mathieu_group_M22
Binary relation over a set and itself
containing R. Reflexive reduction, R≠ Defined as R≠ = R \ {(x, x) | x ∈ X} or the largest irreflexive relation over X contained in R. Transitive closure, R+ Defined
Homogeneous_relation
Type of grammatical voice
reflexive and reciprocal constructions, the unifying feature being a reduction in transitivity. Indeed, it is more common for languages to have a coincidence
Antipassive_voice
Relation specifying a rewrite for each object, compatible with a reduction relation
{\overset {+}{\to }}} , where → + {\displaystyle {\overset {+}{\to }}} is the transitive closure of → {\displaystyle \to } (but not the reflexive closure). In
Reduction_strategy
Open problem in probability theory
{\displaystyle n} . The conjecture has been verified for a large class of transitive graphs whose growth is faster than any polynomial. The foundational result
Dying_percolation_conjecture
Siouan language in Montana
stative verbs, active transitive verbs, and from active intransitive verbs. ii – 'instrumental nominalizer': Derived from active transitive and intransitive
Crow_language
Endangered Salishan language of the US
transitivizer used in the predicate, those occurring with m primarily occur with the causative transitivizer -st(u)- while all other transitivizers take
Coeur_d'Alene_language
Cariban language of Brazil, Suriname and Guyana
the subject of a transitive clause is marked with the postposition _:ja; the subjects of intransitive clauses and objects of transitive sentences are both
Tiriyó_language
Property of rewriting systems in mathematics
Since a sequence of reduction sequences is again a reduction sequence (or, equivalently, since forming the reflexive-transitive closure is idempotent)
Confluence (abstract rewriting)
Confluence_(abstract_rewriting)
Mayan language of Chiapas, Mexico
classifiers. First, some transitive roots reduce valence by infixing -j- into the root. This process is accompanied by a reduction of the number of core
Chʼol_language
Verb whose direct object is the same as its subject
return.REFL+NPST When will you return? The same valence-reduction process occurs for the transitive wagil 'cut' Gaari NEG wagi-iyi cut-REFL+IMP Gaari wagi-iyi
Reflexive_verb
Language family of the Arctic and sub-Arctic
intransitive verbs and objects of transitive verbs are marked with the absolutive case, while subjects of transitive verbs are marked with the ergative
Eskaleut_languages
Class of computational complexity
logic with the addition of a transitive closure operator. A full transitive closure is not needed; a commutative transitive closure and even weaker forms
PSPACE
Order whose elements are all comparable
and b ≤ c {\displaystyle b\leq c} then a ≤ c {\displaystyle a\leq c} (transitive). If a ≤ b {\displaystyle a\leq b} and b ≤ a {\displaystyle b\leq a} then
Total_order
Sporadic simple group
groups and was introduced by Mathieu (1861, 1873). It is a sharply 5-transitive permutation group on 12 objects. Burgoyne & Fong (1968) showed that the
Mathieu_group_M12
Set theory concept
{\displaystyle X} is a preorder ≤ {\displaystyle \leq } on X {\displaystyle X} (a transitive and reflexive relation on X {\displaystyle X} ) that is strongly connected
Prewellordering
Sporadic simple group
sporadic groups and was introduced by Mathieu (1861, 1873). It is a 5-transitive permutation group on 24 objects. The Schur multiplier and the outer automorphism
Mathieu_group_M24
Relation between state transition systems in computer science
also closed under reflexive and transitive closure; therefore, the largest simulation must be reflexive and transitive. From this follows that the largest
Simulation_(computer_science)
Munda language of South Asia
Medio-passive voice. Transitive roots, transitive-intransitive roots, and causative stems will take -ok to derive passive stems. In the transitive-intransitive
Santali_language
Group whose operation is composition of permutations
permutations of {1, 2, 3, 4} is not transitive (no group element takes 1 to 3) but the group of symmetries of a square is transitive on the vertices. A permutation
Permutation_group
Roof of a building that is designed to provide temporary water storage
utilize a novel passive blue roof tray design which relies on the lateral transitivity of non-woven filter fabric for drawdown control in a full scale pilot
Blue_roof
Graph that can be embedded in the plane
ISBN 978-3-540-12687-4. Halldórsson, M.; Kitaev, S.; Pyatkin., A. (2016), "Semi-transitive orientations and word-representable graphs" (PDF), Discr. Appl. Math.
Planar_graph
Pattern in mathematics and computer science
In mathematics and theoretical computer science, a pattern is an unavoidable pattern if it is unavoidable on any finite alphabet. Like a word, a pattern
Unavoidable_pattern
Arabic-language term for a truce or armistice
Ibn Manzur defined it as: "hadana: he grew quiet. hadina: he quieted (transitive or intransitive). haadana: he made peace with. The noun from each of these
Hudna
Vocabulary of a language or branch of knowledge
language's rules. For example, the suffix "-able" is usually only added to transitive verbs, as in "readable" but not "cryable". A compound word is a lexeme
Lexicon
acyclic graph Directed graph Distance regular graph Distance-transitive graph Edge-transitive graph Interval graph Interval graph, improper Interval graph
List_of_graph_theory_topics
Sporadic simple group
and the outer automorphism group are both trivial. M11 is a sharply 4-transitive permutation group on 11 objects. It admits many generating sets of permutations
Mathieu_group_M11
Language of the Yupik family
grammatical case (the absolutive) as the objects of transitive verbs, while the subjects of transitive verbs have a different case (the ergative). For example
Central_Alaskan_Yupʼik
Country in South Asia
original on 17 July 2011. Retrieved 17 July 2011. Lowe, John J. (2017). Transitive Nouns and Adjectives: Evidence from Early Indo-Aryan. Oxford University
India
Grammatical form
exceptional case-marking. As shown in the above examples, the object of the transitive verb want and the preposition for allude to their respective pronouns'
Infinitive
Total order in computer science
ordering to another one, and satisfying the following properties: If (>) is transitive, then so is O(>). If (>) is irreflexive, then so is O(>). If s > t, then
Path ordering (term rewriting)
Path_ordering_(term_rewriting)
Reduction of data redundancy
the Book and Price tables conform to 2NF. The Book table still has a transitive functional dependency ({Author Nationality} is dependent on {Author},
Database_normalization
Reconstructed ancestor of the Circassian languages
action) While most transitive verbs in Circassian are inherently dynamic, the verb Iыгъын (to hold) is a unique bivalent transitive static verb. In its
Proto-Circassian_language
West Germanic language
or direct object of a transitive verb), and of the Old English dative case (for a recipient or indirect object of a transitive verb). The subjective is
English_language
Indigenous language of South America
which take chendal. Intransitive verbs can take either conjugation, transitive verbs normally take areal, but can take chendal for habitual readings
Guarani_language
Samoyedic language
constraint on word order. Below are examples of the basic word order for a transitive and intransitive sentence. məy°mpə-da cheerful-IMPF.PART Wera Wera Maša-m
Tundra_Nenets_language
enumeration It also includes an ILP solver based on generalized basis reduction, transitive closures on maps (which may encode infinite graphs), dependence
Integer_set_library
Fiber bundle whose fibers are group torsors
to the base space X {\displaystyle X} . Because the action is free and transitive, the fibers have the structure of G-torsors. A G {\displaystyle G} -torsor
Principal_bundle
Order by which an agent ranks alternatives based on their utility
option that maximizes their self-interest. But preferences are not always transitive, both because real humans are far from always being rational and because
Preference_(economics)
American mathematician (1936–2024)
Cartan's papers from the early 1900s on the classification of the simple transitive infinite Lie pseudogroups, and of relating Cartan's results to recent
Shlomo_Sternberg
Ways how entities stand to each other
propositions while causal relations connect concrete events. Symmetric, transitive, and reflexive relations are distinguished by their structural features
Relation_(philosophy)
Complexity class
standard solution returned by A 1 {\displaystyle A_{1}} . PLS-reductions are transitive. The transition graph T I {\displaystyle T_{I}} of an instance
PLS_(complexity)
English philosopher (1944–2014)
Stressing the need to retain both the subjective epistemological, or 'transitive', side of knowledge and the objective ontological, or 'intransitive',
Roy_Bhaskar
Concept in mathematics
In mathematics, a Frobenius group is a transitive permutation group on a finite set, such that no non-trivial element fixes more than one point and some
Frobenius_group
Linear representation in mathematics
Steinberg representation. Some of the sporadic simple groups act as doubly transitive permutation groups so have a BN-pair for which one can define a Steinberg
Steinberg_representation
Generalization of graph theory
hypergraphs; Hall-type theorems for hypergraphs. In directed hypergraphs: transitive closure, and shortest path problems. Although hypergraphs are more difficult
Hypergraph
Language of the ancient Urartu, now the Eastern Anatolia region
and the object of a transitive verb are expressed identically, with the so-called absolutive case, whereas the subject of a transitive verb is expressed
Urartian_language
Concept in group theory
of G on the cosets of B is doubly transitive. So (B, N) pairs of rank 1 are more or less the same as doubly transitive actions on sets with more than 2
(B,_N)_pair
Construction in group theory
sharply 3-transitive (faithful and 3-transitive), so the map is one-to-one and has image a 3-transitive subgroup. Thus the image is a 3-transitive subgroup
Projective_linear_group
Sporadic simple group
sporadic groups and was introduced by Mathieu (1861, 1873). It is a 4-fold transitive permutation group on 23 objects. The Schur multiplier and the outer automorphism
Mathieu_group_M23
Problem in computer science
There are two main types of reductions used in computational complexity theory, the many-one reduction and the Turing reduction. A set is many-one reducible
Halting_problem
Logical quantifier
5. {\displaystyle a+2=5{\text{ and }}b+2=5.} Then since equality is a transitive relation, a + 2 = b + 2. {\displaystyle a+2=b+2.} Subtracting 2 from both
Uniqueness_quantification
Existence of values making formula true
sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter Recursive Fuzzy Universal Universe constructible Grothendieck
Satisfiability
English words that indicate a question is being asked, as a grammatical category
underwent further sound changes and spelling changes, notably wh-cluster reductions, resulting in the initial sound being either /w/ (in most dialects) or
English_interrogative_words
English philosopher and logician (1872–1970)
Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset · Superset Transitive Uncountable Universal Theories Alternative Axiomatic Constructive Naive
Bertrand_Russell
Algorithm for shuffling a finite sequence
sorting algorithm may depend on properties of the order relation (like transitivity) that a comparison producing random values will certainly not have. While
Fisher–Yates_shuffle
Extinct language family
suffixes of the head. In verbs, the type of valency, intransitive vs transitive, is signalled by a special suffix, the so-called "class marker". The complex
Hurro-Urartian_languages
Earliest stage of the German language
showed case and gender endings—for intransitive verbs the nominative, for transitive verbs the accusative. For example: After thie thö argangana warun ahtu
Old_High_German
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