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109 relations: Absorption law, Abstract algebra, Abstract interpretation, Algebraic structure, American Mathematical Society, Analogical modeling, Annals of Mathematics, Associative property, Atom (order theory), Automated reasoning, Axiom, Bijection, Binary operation, Boolean algebra (structure), Cambridge University Press, Cartesian product, Category theory, Closure operator, Commutative property, Compact element, Complemented lattice, Complete Heyting algebra, Complete lattice, Completely distributive lattice, Completeness (order theory), Covering relation, Directed set, Distributive property, Distributivity (order theory), Divisibility rule, Domain theory, Duality (order theory), Eulerian poset, Filter (mathematics), Formal concept analysis, Fuzzy set, Galois connection, Garrett Birkhoff, Graded poset, Greatest and least elements, Greatest common divisor, Group (mathematics), Heyting algebra, Homomorphism, Ideal (order theory), Ideal (ring theory), Idempotence, Identity (mathematics), Identity element, Infimum and supremum, ..., Information flow, Intersection (set theory), Invariant subspace, Inverse function, Isomorphism, Join and meet, Knowledge space, Lattice (discrete subgroup), Lattice of subgroups, Least common multiple, Limit-preserving function (order theory), Magma (algebra), Map of lattices, Mathematical Association of America, Mathematical induction, Mathematical maturity, Mathematics, Median graph, Module (mathematics), Monoid, Monotonic function, Morphism, Multiple inheritance, Natural number, Negation, Normal subgroup, Ontology (information science), Operation (mathematics), Order theory, Ordinal optimization, Partial function, Partially ordered set, Partition of a set, Philip M. Whitman, Pointless topology, Post's lattice, Power set, Quantum logic, Ranked poset, Real number, Ring (mathematics), Robert P. Dilworth, Scott information system, Semantics (computer science), Semigroup, Semilattice, Semimodular lattice, Skew lattice, Spectral space, Subset, Subsumption lattice, Tamari lattice, Total order, Union (set theory), Universal algebra, Universal property, Vacuous truth, Young–Fibonacci lattice, 0,1-simple lattice. Expand index (59 more) »
Absorption law
In algebra, the absorption law or absorption identity is an identity linking a pair of binary operations.
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Abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures.
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Abstract interpretation
In computer science, abstract interpretation is a theory of sound approximation of the semantics of computer programs, based on monotonic functions over ordered sets, especially lattices.
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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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American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs.
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Analogical modeling
Analogical modeling (AM) is a formal theory of exemplar based analogical reasoning, proposed by Royal Skousen, professor of Linguistics and English language at Brigham Young University in Provo, Utah.
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Annals of Mathematics
The Annals of Mathematics is a bimonthly mathematical journal published by Princeton University and the Institute for Advanced Study.
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Associative property
In mathematics, the associative property is a property of some binary operations.
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Atom (order theory)
In the mathematical field of order theory, an element a of a partially ordered set with least element 0 is an atom if 0 0 has an atom a below it, that is, there is some a such that b ≥ a:> 0.
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Automated reasoning
Automated reasoning is an area of computer science and mathematical logic dedicated to understanding different aspects of reasoning.
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Axiom
An axiom or postulate is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments.
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Bijection
In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set.
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Binary operation
In mathematics, a binary operation on a set is a calculation that combines two elements of the set (called operands) to produce another element of the set.
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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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Cambridge University Press
Cambridge University Press (CUP) is the publishing business of the University of Cambridge.
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Cartesian product
In set theory (and, usually, in other parts of mathematics), a Cartesian product is a mathematical operation that returns a set (or product set or simply product) from multiple sets.
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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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Closure operator
In mathematics, a closure operator on a set S is a function \operatorname: \mathcal(S)\rightarrow \mathcal(S) from the power set of S to itself which satisfies the following conditions for all sets X,Y\subseteq S |- | X \subseteq \operatorname(X) | (cl is extensive) |- | X\subseteq Y \Rightarrow \operatorname(X) \subseteq \operatorname(Y) | (cl is increasing) |- | \operatorname(\operatorname(X)).
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Commutative property
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result.
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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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Complemented lattice
In the mathematical discipline of order theory, a complemented lattice is a bounded lattice (with least element 0 and greatest element 1), in which every element a has a complement, i.e. an element b satisfying a ∨ b.
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Complete Heyting algebra
In mathematics, especially in order theory, a complete Heyting algebra is a Heyting algebra that is complete as a lattice.
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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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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 (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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Covering relation
In mathematics, especially order theory, the covering relation of a partially ordered set is the binary relation which holds between comparable elements that are immediate neighbours.
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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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Distributive property
In abstract algebra and formal logic, the distributive property of binary operations generalizes the distributive law from boolean algebra and elementary algebra.
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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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Divisibility rule
A divisibility rule is a shorthand way of determining whether a given integer is divisible by a fixed divisor without performing the division, usually by examining its digits.
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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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Eulerian poset
In combinatorial mathematics, an Eulerian poset is a graded poset in which every nontrivial interval has the same number of elements of even rank as of odd rank.
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Filter (mathematics)
In mathematics, a filter is a special subset of a partially ordered set.
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Formal concept analysis
Formal concept analysis (FCA) is a principled way of deriving a concept hierarchy or formal ontology from a collection of objects and their properties.
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Fuzzy set
In mathematics, fuzzy sets (aka uncertain sets) are somewhat like sets whose elements have degrees of membership.
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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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Garrett Birkhoff
Garrett Birkhoff (January 19, 1911 – November 22, 1996) was an American mathematician.
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Graded poset
In mathematics, in the branch of combinatorics, a graded poset is a partially ordered set (poset) P equipped with a rank function ρ from P to N satisfying the following two properties.
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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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Greatest common divisor
In mathematics, the greatest common divisor (gcd) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers.
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Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.
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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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Homomorphism
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces).
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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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Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring.
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Idempotence
Idempotence is the property of certain operations in mathematics and computer science that they can be applied multiple times without changing the result beyond the initial application.
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Identity (mathematics)
In mathematics an identity is an equality relation A.
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Identity element
In mathematics, an identity element or neutral element is a special type of element of a set with respect to a binary operation on that set, which leaves other elements unchanged when combined with them.
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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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Information flow
In discourse-based grammatical theory, information flow is any tracking of referential information by speakers.
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Intersection (set theory)
In mathematics, the intersection A ∩ B of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements.
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Invariant subspace
In mathematics, an invariant subspace of a linear mapping T: V → V from some vector space V to itself is a subspace W of V that is preserved by T; that is, T(W) ⊆ W.
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Inverse function
In mathematics, an inverse function (or anti-function) is a function that "reverses" another function: if the function applied to an input gives a result of, then applying its inverse function to gives the result, and vice versa.
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Isomorphism
In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that can be reversed by an inverse morphism.
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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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Knowledge space
In mathematical psychology, a knowledge space is a combinatorial structure describing the possible states of knowledge of a human learner.
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Lattice (discrete subgroup)
In Lie theory and related areas of mathematics, a lattice in a locally compact group is a discrete subgroup with the property that the quotient space has finite invariant measure.
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Lattice of subgroups
In mathematics, the lattice of subgroups of a group G is the lattice whose elements are the subgroups of G, with the partial order relation being set inclusion.
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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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Magma (algebra)
In abstract algebra, a magma (or groupoid; not to be confused with groupoids in category theory) is a basic kind of algebraic structure.
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Map of lattices
The concept of a lattice arises in order theory, a branch of mathematics.
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Mathematical Association of America
The Mathematical Association of America (MAA) is a professional society that focuses on mathematics accessible at the undergraduate level.
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Mathematical induction
Mathematical induction is a mathematical proof technique.
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Mathematical maturity
Mathematical maturity is an informal term used by mathematicians to refer to a mixture of mathematical experience and insight that cannot be directly taught.
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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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Median graph
In graph theory, a division of mathematics, a median graph is an undirected graph in which every three vertices a, b, and c have a unique median: a vertex m(a,b,c) that belongs to shortest paths between each pair of a, b, and c. The concept of median graphs has long been studied, for instance by or (more explicitly) by, but the first paper to call them "median graphs" appears to be.
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Module (mathematics)
In mathematics, a module is one of the fundamental algebraic structures used in abstract algebra.
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Monoid
In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element.
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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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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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Multiple inheritance
Multiple inheritance is a feature of some object-oriented computer programming languages in which an object or class can inherit characteristics and features from more than one parent object or parent class.
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Natural number
In mathematics, the natural numbers are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country").
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Negation
In logic, negation, also called the logical complement, is an operation that takes a proposition P to another proposition "not P", written \neg P (¬P), which is interpreted intuitively as being true when P is false, and false when P is true.
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Normal subgroup
In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group of which it is a part.
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Ontology (information science)
In computer science and information science, an ontology encompasses a representation, formal naming, and definition of the categories, properties, and relations of the concepts, data, and entities that substantiate one, many, or all domains.
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Operation (mathematics)
In mathematics, an operation is a calculation from zero or more input values (called operands) to an output value.
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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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Ordinal optimization
In mathematical optimization, ordinal optimization is the maximization of functions taking values in a partially ordered set ("poset").
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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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Partition of a set
In mathematics, a partition of a set is a grouping of the set's elements into non-empty subsets, in such a way that every element is included in one and only one of the subsets.
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Philip M. Whitman
Philip Martin Whitman is an American mathematician who contributed to lattice theory, in particular to the theory of free lattices.
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Pointless topology
In mathematics, pointless topology (also called point-free or pointfree topology, or locale theory) is an approach to topology that avoids mentioning points.
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Post's lattice
In logic and universal algebra, Post's lattice denotes the lattice of all clones on a two-element set, ordered by inclusion.
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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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Quantum logic
In quantum mechanics, quantum logic is a set of rules for reasoning about propositions that takes the principles of quantum theory into account.
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Ranked poset
In mathematics, a ranked partially ordered set - or poset - may be either.
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Real number
In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.
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Ring (mathematics)
In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra.
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Robert P. Dilworth
Robert Palmer Dilworth (December 2, 1914 – October 29, 1993) was an American mathematician.
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Scott information system
In domain theory, a branch of mathematics and computer science, a Scott information system is a primitive kind of logical deductive system often used as an alternative way of presenting Scott domains.
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Semantics (computer science)
In programming language theory, semantics is the field concerned with the rigorous mathematical study of the meaning of programming languages.
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Semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation.
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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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Semimodular lattice
In the branch of mathematics known as order theory, a semimodular lattice, is a lattice that satisfies the following condition:;Semimodular law: a ∧ b These definitions follow Stern (1999).
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Skew lattice
In abstract algebra, a skew lattice is an algebraic structure that is a non-commutative generalization of a lattice.
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Spectral space
In mathematics, a spectral space (sometimes called a coherent space) is a topological space that is homeomorphic to the spectrum of a commutative ring.
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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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Subsumption lattice
A subsumption lattice is a mathematical structure used in theoretical background of automated theorem proving and other symbolic computation applications.
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Tamari lattice
In mathematics, a Tamari lattice, introduced by, is a partially ordered set in which the elements consist of different ways of grouping a sequence of objects into pairs using parentheses; for instance, for a sequence of four objects abcd, the five possible groupings are ((ab)c)d, (ab)(cd), (a(bc))d, a((bc)d), and a(b(cd)).
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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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Universal property
In various branches of mathematics, a useful construction is often viewed as the “most efficient solution” to a certain problem.
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Vacuous truth
In mathematics and logic, a vacuous truth is a statement that asserts that all members of the empty set have a certain property.
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Young–Fibonacci lattice
In mathematics, the Young–Fibonacci graph and Young–Fibonacci lattice, named after Alfred Young and Leonardo Fibonacci, are two closely related structures involving sequences of the digits 1 and 2.
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0,1-simple lattice
In lattice theory, a bounded lattice L is called a 0,1-simple lattice if nonconstant lattice homomorphisms of L preserve the identity of its top and bottom elements.
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Bounded lattice, Complement (order theory), Conditionally complete lattice, Continuous lattice, Join (lattice theory), Join-irreducible, Join-prime, Jordan-Dedekind chain condition, Jordan-Dedekind lattice, Jordan-Dedekind property, Jordan-dedekind lattice, Jordan-dedekind property, Jordan–Dedekind chain condition, Jordan–Dedekind lattice, Lattice (algebra), Lattice (order theory), Lattice Automorphism, Lattice Endomorphism, Lattice Homomorphism, Lattice Isomorphism, Lattice automorphism, Lattice endomorphism, Lattice homomorphism, Lattice isomorphism, Lattice order, Lattice theory, Meet (lattice theory), Meet-irreducible, Meet-prime, Partial lattice, Sublattice.
References
[1] https://en.wikipedia.org/wiki/Lattice_(order)
