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40 relations: Algebraic structure, Axiom of choice, Boolean algebra (structure), Bounded complete poset, Category theory, Chain-complete partial order, Complete lattice, Complete partial order, Completely distributive lattice, Completeness of the real numbers, Dedekind cut, Directed set, Distributivity (order theory), Domain theory, Duality (order theory), Empty set, Galois connection, Greatest and least elements, Heyting algebra, Ideal (order theory), Identity (mathematics), If and only if, Infimum and supremum, Join and meet, Lattice (order), Least common multiple, Limit-preserving function (order theory), Mathematical induction, Mathematics, Morphism, Order theory, Partially ordered set, Power set, Product order, Semilattice, Subset, Total order, Union (set theory), Universal algebra, Upper set.
Algebraic structure
In mathematics, and more specifically in abstract algebra, an algebraic structure on a set A (called carrier set or underlying set) is a collection of finitary operations on A; the set A with this structure is also called an algebra.
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Axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that the Cartesian product of a collection of non-empty sets is non-empty.
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Boolean algebra (structure)
In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice.
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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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Category theory
Category theory formalizes mathematical structure and its concepts in terms of a labeled directed graph called a category, whose nodes are called objects, and whose labelled directed edges are called arrows (or morphisms).
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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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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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Complete partial order
In mathematics, the phrase complete partial order is variously used to refer to at least three similar, but distinct, classes of partially ordered sets, characterized by particular completeness properties.
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Completely distributive lattice
In the mathematical area of order theory, a completely distributive lattice is a complete lattice in which arbitrary joins distribute over arbitrary meets.
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Completeness of the real numbers
Intuitively, completeness implies that there are not any “gaps” (in Dedekind's terminology) or “missing points” in the real number line.
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Dedekind cut
In mathematics, Dedekind cuts, named after German mathematician Richard Dedekind, are а method of construction of the real numbers from the rational numbers.
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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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Distributivity (order theory)
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.
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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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Galois connection
In mathematics, especially in order theory, a Galois connection is a particular correspondence (typically) between two partially ordered sets (posets).
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Greatest and least elements
In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined dually, that is, it is an element of S that is smaller than every other element of S. Formally, given a partially ordered set (P, ≤), an element g of a subset S of P is the greatest element of S if Hence, the greatest element of S is an upper bound of S that is contained within this subset.
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Heyting algebra
In mathematics, a Heyting algebra is a bounded lattice (with join and meet operations written ∨ and ∧ and with least element 0 and greatest element 1) equipped with a binary operation a → b of implication such that c ∧ a ≤ b is equivalent to c ≤ a → b. From a logical standpoint, A → B is by this definition the weakest proposition for which modus ponens, the inference rule A → B, A ⊢ B, is sound.
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Ideal (order theory)
In mathematical order theory, an ideal is a special subset of a partially ordered set (poset).
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Identity (mathematics)
In mathematics an identity is an equality relation A.
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If and only if
In logic and related fields such as mathematics and philosophy, if and only if (shortened iff) is a biconditional logical connective between statements.
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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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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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Lattice (order)
A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra.
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Least common multiple
In arithmetic and number theory, the least common multiple, lowest common multiple, or smallest common multiple of two integers a and b, usually denoted by LCM(a, b), is the smallest positive integer that is divisible by both a and b. Since division of integers by zero is undefined, this definition has meaning only if a and b are both different from zero.
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Limit-preserving function (order theory)
In the mathematical area of order theory, one often speaks about functions that preserve certain limits, i.e. certain suprema or infima.
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Mathematical induction
Mathematical induction is a mathematical proof technique.
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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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Morphism
In mathematics, a morphism is a structure-preserving map from one mathematical structure to another one of the same type.
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Order theory
Order theory is a branch of mathematics which investigates the intuitive notion of order using binary relations.
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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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Power set
In mathematics, the power set (or powerset) of any set is the set of all subsets of, including the empty set and itself, variously denoted as, 𝒫(), ℘() (using the "Weierstrass p"),,, or, identifying the powerset of with the set of all functions from to a given set of two elements,.
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Product order
In mathematics, given two ordered sets A and B, one can induce a partial ordering on the Cartesian product.
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Semilattice
In mathematics, a join-semilattice (or upper semilattice) is a partially ordered set that has a join (a least upper bound) for any nonempty finite subset.
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Subset
In mathematics, a set A is a subset of a set B, or equivalently B is a superset of A, if A is "contained" inside B, that is, all elements of A are also elements of B. A and B may coincide.
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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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Union (set theory)
In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection.
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Universal algebra
Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures.
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Upper set
In mathematics, an upper set (also called an upward closed set or just an upset) of a partially ordered set (X,≤) is a subset U with the property that, if x is in U and x≤y, then y is in U. The dual notion is lower set (alternatively, down set, decreasing set, initial segment, semi-ideal; the set is downward closed), which is a subset L with the property that, if x is in L and y≤x, then y is in L. The terms order ideal or ideal are sometimes used as synonyms for lower set.
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Redirects here:
Completeness axiom, Completeness properties, Completeness property.
References
[1] https://en.wikipedia.org/wiki/Completeness_(order_theory)
