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31 relations: Bounded complete poset, Cartesian closed category, Chain-complete partial order, Compact element, Complete lattice, Completeness (order theory), Denotational semantics, Directed set, Domain theory, Duality (order theory), Empty set, Filter (mathematics), Fixed point (mathematics), Formal system, Graph of a function, Henk Barendregt, Infimum and supremum, Iterated function, Join and meet, Kleene fixed-point theorem, Knaster–Tarski theorem, Mathematics, Monotonic function, North Holland, Partial function, Partially ordered set, Scott continuity, Sober space, Specialization (pre)order, Theoretical computer science, Total order.
Bounded complete poset
In the mathematical field of order theory, a partially ordered set is bounded complete if all of its subsets that have some upper bound also have a least upper bound.
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Cartesian closed category
In category theory, a category is considered Cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors.
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Chain-complete partial order
In order-theoretic mathematics, a partially ordered set is chain-complete if every chain in it has a least upper bound.
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Compact element
In the mathematical area of order theory, the compact or finite elements of a partially ordered set are those elements that cannot be subsumed by a supremum of any non-empty directed set that does not already contain members above the compact element.
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Complete lattice
In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet).
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Completeness (order theory)
In the mathematical area of order theory, completeness properties assert the existence of certain infima or suprema of a given partially ordered set (poset).
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Denotational semantics
In computer science, denotational semantics (initially known as mathematical semantics or Scott–Strachey semantics) is an approach of formalizing the meanings of programming languages by constructing mathematical objects (called denotations) that describe the meanings of expressions from the languages.
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Directed set
In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive and transitive binary relation ≤ (that is, a preorder), with the additional property that every pair of elements has an upper bound.
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Domain theory
Domain theory is a branch of mathematics that studies special kinds of partially ordered sets (posets) commonly called domains.
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Duality (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 by Pop or Pd.
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Empty set
In mathematics, and more specifically set theory, the empty set or null set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero.
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Filter (mathematics)
In mathematics, a filter is a special subset of a partially ordered set.
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Fixed point (mathematics)
In mathematics, a fixed point (sometimes shortened to fixpoint, also known as an invariant point) of a function is an element of the function's domain that is mapped to itself by the function.
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Formal system
A formal system is the name of a logic system usually defined in the mathematical way.
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Graph of a function
In mathematics, the graph of a function f is, formally, the set of all ordered pairs, and, in practice, the graphical representation of this set.
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Henk Barendregt
Hendrik Pieter (Henk) Barendregt (born 18 December 1947, Amsterdam) is a Dutch logician, known for his work in lambda calculus and type theory.
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Infimum and supremum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set T is the greatest element in T that is less than or equal to all elements of S, if such an element exists.
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Iterated function
In mathematics, an iterated function is a function (that is, a function from some set to itself) which is obtained by composing another function with itself a certain number of times.
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Join and meet
In a partially ordered set P, the join and meet of a subset S are respectively the supremum (least upper bound) of S, denoted ⋁S, and infimum (greatest lower bound) of S, denoted ⋀S.
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Kleene fixed-point theorem
In the mathematical areas of order and lattice theory, the Kleene fixed-point theorem, named after American mathematician Stephen Cole Kleene, states the following: The ascending Kleene chain of f is the chain obtained by iterating f on the least element ⊥ of L. Expressed in a formula, the theorem states that where \textrm denotes the least fixed point.
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Knaster–Tarski theorem
In the mathematical areas of order and lattice theory, the Knaster–Tarski theorem, named after Bronisław Knaster and Alfred Tarski, states the following: It was Tarski who stated the result in its most general form, and so the theorem is often known as Tarski's fixed point theorem.
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Mathematics
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
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Monotonic function
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order.
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North Holland
North Holland (Noord-Holland, West Frisian Dutch: Noard-Holland) is a province of the Netherlands located in the northwestern part of the country.
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Partial function
In mathematics, a partial function from X to Y (written as or) is a function, for some subset X ′ of X.
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Partially ordered set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set.
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Scott continuity
In mathematics, given two partially ordered sets P and Q, a function f \colon P \rightarrow Q between them is Scott-continuous (named after the mathematician Dana Scott) if it preserves all directed suprema, i.e. if for every directed subset D of P with supremum in P its image has a supremum in Q, and that supremum is the image of the supremum of D: that is, \sqcup f.
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Sober space
In mathematics, a sober space is a topological space X such that every irreducible closed subset of X is the closure of exactly one point of X: that is, this closed subset has a unique generic point.
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Specialization (pre)order
In the branch of mathematics known as topology, the specialization (or canonical) preorder is a natural preorder on the set of the points of a topological space.
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Theoretical computer science
Theoretical computer science, or TCS, is a subset of general computer science and mathematics that focuses on more mathematical topics of computing and includes the theory of computation.
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Total order
In mathematics, a linear order, total order, simple order, or (non-strict) ordering is a binary relation on some set X, which is antisymmetric, transitive, and a connex relation.
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Redirects here:
Complete poset, Directed complete partial order.
References
[1] https://en.wikipedia.org/wiki/Complete_partial_order
