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In the mathematical area of order theory, there are various notions of the common concept of distributivity, applied to the formation of suprema and infima
Distributivity_(order_theory)
Property involving two mathematical operations
is a distinction between left-distributivity and right-distributivity: a ⋅ ( b ± c ) = a ⋅ b ± a ⋅ c (left-distributive) {\displaystyle a\cdot \left(b\pm
Distributive_property
Special type of lattice
conditions of order theory can be found in the article Distributivity (order theory). Sholander (1951) gives a simplified characterization of distributive lattices
Distributive_lattice
Existence of certain infima or suprema of a given poset
constructively completely distributive. See also the articles on complete distributivity and distributivity (order theory). The considerations in this
Completeness_(order_theory)
Branch of mathematics
distributivity laws which are discussed in the article on distributivity in order theory. Some additional order structures that are often specified via algebraic
Order_theory
Set whose pairs have minima and maxima
as frames and completely distributive lattices, see distributivity in order theory. For some applications the distributivity condition is too strong,
Lattice_(order)
Ultrafilter Completeness (order theory) Dense order Distributivity (order theory) Modular lattice Distributive lattice Completely distributive lattice Ascending
List_of_order_theory_topics
Term in the mathematical area of order theory
In the mathematical area of order theory, every partially ordered set P gives rise to a dual (or opposite) partially ordered set which is often denoted
Duality_(order_theory)
Order whose elements are all comparable
Lattice theory: first concepts and distributive lattices. W. H. Freeman and Co. ISBN 0-7167-0442-0 Halmos, Paul R. (1968). Naive Set Theory. Princeton:
Total_order
Generalized alphabetical order
lexicographic or lexicographical order (also known as lexical order, or dictionary order) is a generalization of the alphabetical order of the dictionaries to sequences
Lexicographic_order
Mathematical set with an ordering
In mathematics, especially order theory, a partial order on a set is an arrangement such that, for certain pairs of elements, one precedes the other.
Partially_ordered_set
Type of logical system
first-order logic is an extension of propositional logic. A theory about a topic, such as set theory, a theory for groups, or a formal theory of arithmetic
First-order_logic
Partial order with joins
binary meets exist is a distributive lattice. See the entry distributivity (order theory). A join-semilattice is distributive if and only if the lattice
Semilattice
1971 book by John Rawls
author attempts to provide a moral theory alternative to utilitarianism and that addresses the problem of distributive justice (the socially just distribution
A_Theory_of_Justice
In the mathematical area of order theory, a completely distributive lattice is a complete lattice in which arbitrary joins distribute over arbitrary meets
Completely distributive lattice
Completely_distributive_lattice
Concept relating to distribution of rewards to group members
distribution of benefits and burdens within a society. Most contemporary theories of distributive justice rest on the precondition of material scarcity. From that
Distributive_justice
Generalized object in category theory
In category theory, the product of two (or more) objects in a category is a notion designed to capture the essence behind constructions in other areas
Product_(category_theory)
Order-preserving mathematical function
reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory. In calculus, a function
Monotonic_function
Equivalence of distributive lattices and set families
duality theory for distributive lattices. Birkhoff's representation theorem may also be generalized to finite structures other than distributive lattices
Birkhoff's representation theorem
Birkhoff's_representation_theorem
Nonempty, upper-bounded, downward-closed subset
In mathematical order theory, an ideal is a special subset of a partially ordered set (poset). Although this term historically was derived from the notion
Ideal_(order_theory)
Subset of incomparable elements
In mathematics, in the area of order theory, an antichain is a subset of a partially ordered set such that any two distinct elements in the subset are
Antichain
Theories in mathematical logic
In first-order logic, a first-order theory is given by a set of axioms in some language. This entry lists some of the more common examples used in model
List_of_first-order_theories
Vector space equipped with a bilinear product
space is commutative, left distributivity and right distributivity are equivalent, and, in this case, only one distributivity requires a proof. In general
Algebra_over_a_field
Certain topology in mathematics
is called orderable or linearly orderable if there exists a total order on its elements such that the order topology induced by that order and the given
Order_topology
Equivalence of partially ordered sets
In the mathematical field of order theory, an order isomorphism is a special kind of monotone function that constitutes a suitable notion of isomorphism
Order_isomorphism
Congressional passage. There are several competing theories on the precise definition of distributive politics, as political scientists approach the concept
Distributive_tendency
Concept of moral fairness and administration of the law
promoting order rather than justice. Modern-day Western notions of justice have roots in Christian theology, which largely follows the divine command theory, according
Justice
Theories in cognitive psychology
multiplication to be coordinated as with long multiplication or distributivity. Furthermore, previous theories of stage have confounded the stimulus and response in
Neo-Piagetian theories of cognitive development
Neo-Piagetian_theories_of_cognitive_development
Class of mathematical orderings
In mathematics, a well-order (or well-ordering or well-order relation) on a set S is a total ordering on S with the property that every non-empty subset
Well-order
Study of parts and the wholes they form
Dordrecht: Foris. pp. 75–115. Champollion, Lucas (2017). Parts of a Whole: Distributivity as a Bridge Between Aspect and Measurement. Oxford Studies in Theoretical
Mereology
Equalities for combinations of sets
becomes the innermost operator. Right distributivity: ( L ∩ M ) ∪ R = ( L ∪ R ) ∩ ( M ∪ R ) (Right-distributivity of ∪ over ∩ ) ( L ∪ M ) ∪ R
List of set identities and relations
List_of_set_identities_and_relations
The theory of criminal justice is the branch of philosophy of law that deals with criminal justice and in particular punishment. The theory of criminal
Theory_of_criminal_justice
Alternative mathematical ordering
mathematics, a cyclic order is a way to arrange a set of objects in a circle.[nb] Unlike most structures in order theory, a cyclic order is not modeled as
Cyclic_order
Economic theory promoting local control
Distributism is an economic theory asserting that the world's productive assets should be widely owned rather than concentrated. Developed in the late
Distributism
Concept in behavioral economics, political theory and behavioral sciences
Nudge theory is a concept in behavioral economics, decision making, behavioral policy, social psychology, consumer behavior, and related behavioral sciences
Nudge_theory
Concept in political philosophy
maintenance of the social order. Conceptualized in the Age of Enlightenment, social contractarianism is a core concept of modern theories of constitutionalism
Social_contract
Generalization theory explaining social behaviour regarding society and economics
of social behavior. Homans based his theory on concepts that include equilibration, expectancy, and a distributive justice in dyadic exchanges. Using this
Social_exchange_theory
Algebraic structure
all elements x of P and all subsets S of P, the following infinite distributivity law holds: x ∧ ⋁ s ∈ S s = ⋁ s ∈ S ( x ∧ s ) . {\displaystyle x\land
Complete_Heyting_algebra
Sets with binary operations analogous to the Reidemeister moves used on knot diagrams
each element acts as automorphisms encodes the left and right self-distributivity laws, and also these laws: a ◃ ( b ▹ c ) = ( a ◃ b ) ▹ ( a ◃ c ) ( c
Racks_and_quandles
Generalization of vector spaces from fields to rings
multiplication. Modules are very closely related to the representation theory of groups. They are also one of the central notions of commutative algebra
Module_(mathematics)
Set with operations obeying given axioms
and right-distributive. If the operation ∗ {\displaystyle *} is commutative, left and right distributivity are both equivalent to distributivity. Some common
Algebraic_structure
Algebraic manipulation of "true" and "false"
associativity, commutativity, and absorption laws, distributivity of ∧ over ∨ (or the other distributivity law—one suffices), and the two complement laws
Boolean_algebra
Glossary of terms used in branch of mathematics
overview articles: completeness properties of partial orders distributivity laws of order theory In the following, partial orders will usually just be denoted
Glossary_of_order_theory
Bound lattice in which every element has a complement
b) ∧ c holds. This is weaker than distributivity; e.g. the above-shown lattice M3 is modular, but not distributive. A natural further weakening of this
Complemented_lattice
coordinates are 0 or 1 are exactly the order polytopes. Stable matching polytope, a convex polytope that defines a distributive lattice on its points in a different
Distributive_polytope
Axioms for the natural numbers
the second-order and first-order formulations, as discussed in the section § Peano arithmetic as first-order theory below. If the second-order induction
Peano_axioms
Type of ordering of a set
upper bounds are order-isomorphic. This makes the theory of dense linear orders without bounds an example of an ω-categorical theory where ω is the smallest
Dense_order
Special subset of a partially ordered set
Filters appear in order and lattice theory, but also topology, whence they originate. The notion dual to a filter is an order ideal. Special cases of filters
Filter_(mathematics)
Operation in algebra and mathematics
In category theory, a branch of mathematics, a monad is a triple ( T , η , μ ) {\displaystyle (T,\eta ,\mu )} consisting of a functor T from a category
Monad_(category_theory)
Theory in communications
Expectancy violations theory (EVT) is a theory of communication that analyzes how individuals respond to unanticipated violations of social norms and expectations
Expectancy_violations_theory
Facet theory is a metatheory for the multivariate behavioral sciences that posits that scientific theories and measurements can be advanced by discovering
Facet_theory
Discrete math concept
discrete mathematics, dominance order (synonyms: dominance ordering, majorization order, natural ordering) is a partial order on the set of partitions of
Dominance_order
Mathematical concept for comparing objects
In mathematics, specifically order theory, a well-quasi-ordering or wqo on a set X {\displaystyle X} is a quasi-ordering of X {\displaystyle X} for which
Well-quasi-ordering
On chains and antichains in partial orders
In mathematics, in the areas of order theory and combinatorics, Dilworth's theorem states that, in any finite partially ordered set, the maximum size of
Dilworth's_theorem
enumerable theories cannot be complete 5. Gödel's completeness theorem states that consistent first-order theories have models 𝔥 The distributivity number
Glossary_of_set_theory
Algebraic structure with addition, multiplication, and division
field theory, or by direct computation. For example, A ⋅ (B + A) = A ⋅ I = A, which equals A ⋅ B + A ⋅ A = I + B = A, as required by the distributivity. This
Field_(mathematics)
Concept in order theory
In mathematics, specifically order theory, the join of a subset S {\displaystyle S} of a partially ordered set P {\displaystyle P} is the supremum (least
Join_and_meet
theory, the Dehornoy order is a left-invariant total order on the braid group, found by Patrick Dehornoy. Dehornoy's original discovery of the order on
Dehornoy_order
Type of monotone function
In order theory, a branch of mathematics, an order embedding is a special kind of monotone function, which provides a way to include one partially ordered
Order_embedding
Construction in order theory
Introduction to Lattices and Order (Second Edition), 2002, p. 18 Alexander Shen; Nikolai Konstantinovich Vereshchagin (2002). Basic Set Theory. American Mathematical
Product_order
Set of elements in any of some sets
In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection. It is one of the fundamental operations through
Union_(set_theory)
more specifically in order theory, several different types of ordered set have been studied. They include: Cyclic orders, orderings in which triples of
List of order structures in mathematics
List_of_order_structures_in_mathematics
Political philosophy emphasising social ownership of production
that "as far as economic theory was concerned, there is nothing in principle in economic theory that precludes an economic order based on socialist policies"
Socialism
Legal and philosophical theory that there are values inherent in nature
Natural law (Latin: ius naturale, lex naturalis) is a philosophical and legal theory that posits the existence of inherent laws derived from nature and universal
Natural_law
Theorem in order theory
Dushnik–Miller theorem is a result in order theory stating that every countably infinite linear order has a non-identity order embedding into itself. It is named
Dushnik–Miller_theorem
Sentence structure
assembly" Following the theories of generative grammar, Devine and Stephens assume that deviations from that basic unmarked order are made to put emphasis
Latin_word_order
Standard that diagrams must satisfy up to isomorphism
ISBN 978-1-4612-6900-7. Laplaza, Miguel L. (1972). "Coherence for distributivity". Coherence in Categories. Lecture Notes in Mathematics. Vol. 281. pp
Coherency_(homotopy_theory)
Theory of interpersonal relationships
multiplication and division and the distributive law. The two main, original publications on relational models theory have received over 5000 citations
Relational_models_theory
Social justice theory
organization, is a more extreme outcome stemming from the same equity theory principles. Distributive justice perceptions are most strongly related to withdrawal
Organizational_justice
Isomorphism type of ordered sets
mathematics, especially in set theory and order theory, two ordered sets X and Y are said to have the same order type if they are order isomorphic, that is, if
Order_type
Set of ethical theories
individuals, in order to benefit society as a whole. It sometimes presents as a principle of distributive justice in economic theories (economic limitarianism)
Limitarianism_(ethical)
Set of elements common to all of some sets
In set theory, the intersection of two sets A {\displaystyle A} and B , {\displaystyle B,} denoted by A ∩ B , {\displaystyle A\cap B,} is the set containing
Intersection_(set_theory)
Identities and relationships involving sets
In mathematics, particularly in the study of set theory, the algebra of sets defines the properties and laws of sets, the set-theoretic operations of union
Algebra_of_sets
Mathematical approach
topology (or pointfree topology) or topology without points and locale theory, is an approach to topology where lattices of open sets are the primitive
Pointless_topology
Mathematical ranking of a set
In order theory, a weak ordering is a mathematical formalization of the intuitive notion of a ranking of a set, some of whose members may be tied with
Weak_ordering
Submodule of a mathematical ring
important in number theory). The related, but distinct, concept of an ideal in order theory is derived from the notion of an ideal in ring theory. A fractional
Ideal_(ring_theory)
Theory in social psychology
Social comparison theory, initially proposed by social psychologist Leon Festinger in 1954, centers on the belief that individuals drive to gain accurate
Social_comparison_theory
sets of the partial order, and its dimension is the number of elements in the partial order. The order polytope is a distributive polytope, meaning that
Order_polytope
Arithmetic operation
formalized in the definition of a ring. In some contexts, integers, distributivity over addition, and the existence of a multiplicative identity are enough
Addition
In mathematics, especially abstract algebra, loop theory and quasigroup theory are active research areas with many open problems. As in other areas of
List of problems in loop theory and quasigroup theory
List_of_problems_in_loop_theory_and_quasigroup_theory
Pair of logical equivalences
output, as well as change the operator when doing a substitution. In set theory, it is often stated as "union and intersection interchange under complementation"
De_Morgan's_laws
Algebraic object with an ordered structure
Artin–Schreier theory of ordered fields and formally real fields. There are two equivalent common definitions of an ordered field. The definition of total order appeared
Ordered_field
In order-theoretic mathematics, a series-parallel partial order is a partially ordered set built up from smaller series-parallel partial orders by two
Series-parallel_partial_order
Division ring with weakened conditions
is similarly defined, but satisfies right distributivity instead. A quasifield satisfying both distributive laws is called a semifield, in the sense in
Quasifield
Theory in social science
Face negotiation theory is a theory conceived by Stella Ting-Toomey in 1985, to understand how people from different cultures manage rapport and disagreements
Face_negotiation_theory
In category theory, an allegory is a category that has some of the structure of the category Rel of sets and binary relations between them. Allegories
Allegory_(mathematics)
Ideals in a Boolean algebra can be extended to prime ideals
(of ring theory), or distributive lattices and maximal ideals (of order theory). This article focuses on prime ideal theorems from order theory. Although
Boolean_prime_ideal_theorem
General, formal theory of continuous quantity
The theory of conjoint measurement (also known as conjoint measurement or additive conjoint measurement) is a general, formal theory of continuous quantity
Theory of conjoint measurement
Theory_of_conjoint_measurement
Applying operations to functions in terms of values for each input "point"
operations inherit such properties as associativity, commutativity and distributivity from corresponding operations on the codomain. If A {\displaystyle A}
Pointwise
Arithmetic function
functions", Number theory, Turku: de Gruyter: 115–123 Langford, E. (1973), "Distributivity over the Dirichlet product and completely multiplicative arithmetical
Completely multiplicative function
Completely_multiplicative_function
Laver's theorem, in order theory, states that order embeddability of countable total orders is a well-quasi-ordering. That is, for every infinite sequence
Laver's_theorem
1990 book on mathematical order theory
Introduction to Lattices and Order is a mathematical textbook on order theory by Brian A. Davey and Hilary Priestley. It was published by the Cambridge
Introduction to Lattices and Order
Introduction_to_Lattices_and_Order
Reflexive and transitive binary relation
In mathematics, in particular in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive. The name preorder is meant
Preorder
Algebraic structure with a binary operation
sense used in category theory, but not in the sense used by Hausmann and Ore. Nevertheless, influential books in semigroup theory, including Clifford and
Magma_(algebra)
Property of a mathematical operation
semigroup is a set with an associative binary operation. Commutativity and distributivity are two other frequently discussed properties of binary operations.
Associative_property
Array of numbers
and (A + B)C = AC + BC as well as C(A + B) = CA + CB (left and right distributivity), whenever the size of the matrices is such that the various products
Matrix_(mathematics)
homology theory of a topological space could be defined in terms of its distributive lattice of closed sets. He observed that the inclusion order on the
Disjunction property of Wallman
Disjunction_property_of_Wallman
Generalization of monads
January 2026 (link) Marmolejo, Francisco (1999). "Distributive laws for pseudomonads" (PDF). Theory and Applications of Categories. 5 (5): 81–147. doi:10
Pseudomonad_(category_theory)
Part of speech reflecting the reference of a noun
single, or once (which are considered numerals). Distributive determiners, also called distributive adjectives, consider members of a group separately
Determiner
Algebraic structure
computer science, including number theory, algebraic geometry, Galois theory, finite geometry, cryptography and coding theory. A finite field is a field that
Finite_field
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