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126 relations: Abelian group, Absorbing element, Abstract algebra, Abstract data type, Algebra over a field, Algebraic structure, Associative property, Bicyclic semigroup, Bijection, Binary operation, Binary relation, Binary tree, Boolean algebra (structure), Cambridge University Press, Cancellation property, Cartesian product, Category (mathematics), Category of groups, Category of magmas, Category of sets, Category theory, Character (computing), Class (set theory), Closed manifold, Closure (mathematics), Closure operator, Commutative property, Complex number, Computer science, Concatenation, Concurrent computing, Connected sum, Constant (mathematics), Constant function, Cyclic group, Data structure, Directed set, Empty string, Endomorphism, Equivalence of categories, Exclusive or, Finite-state machine, Fold (higher-order function), Free monoid, Free object, Function composition, Functor, Green's relations, Grothendieck group, Group (mathematics), ..., Group action, Group homomorphism, Heyting algebra, History monoid, Homeomorphism, Homomorphism, Idempotence, Identity element, Identity function, Index set, Infimum and supremum, Integer, Inverse element, Isomorphism, Join and meet, Kleene algebra, Krohn–Rhodes theory, Lattice (order), Lexicographical order, Logical biconditional, Logical conjunction, Logical disjunction, Magma (algebra), MapReduce, Mathematics, Matrix (mathematics), Matrix addition, Matrix multiplication, Monad (functional programming), Monoid, Monoid (category theory), Morphism, Multiset, Natural number, Null semigroup, Numerical semigroup, Operation (mathematics), Oxford University Press, Parallel computing, Partially ordered group, Partially ordered set, Plactic monoid, Pointwise, Power set, Prefix sum, Preorder, Presentation of a group, Process calculus, Product of group subsets, Rational number, Real number, Ring (mathematics), Semiautomaton, Semigroup, Semigroup action, Semilattice, Semiring, Sequence, Set (mathematics), Singleton (mathematics), Springer Science+Business Media, Star height problem, String (computer science), Subset, Syntactic monoid, Torus, Trace monoid, Transformation semigroup, Transition system, Tree traversal, Trivial group, Tuple, Vedic square, Zerosumfree monoid, 0, 1. Expand index (76 more) »
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written.
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Absorbing element
In mathematics, an absorbing element is a special type of element of a set with respect to a binary operation on that set.
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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 data type
In computer science, an abstract data type (ADT) is a mathematical model for data types, where a data type is defined by its behavior (semantics) from the point of view of a user of the data, specifically in terms of possible values, possible operations on data of this type, and the behavior of these operations.
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Algebra over a field
In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product.
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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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Associative property
In mathematics, the associative property is a property of some binary operations.
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Bicyclic semigroup
In mathematics, the bicyclic semigroup is an algebraic object important for the structure theory of semigroups.
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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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Binary relation
In mathematics, a binary relation on a set A is a set of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2.
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Binary tree
In computer science, a binary tree is a tree data structure in which each node has at most two children, which are referred to as the and the.
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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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Cancellation property
In mathematics, the notion of cancellative is a generalization of the notion of invertible.
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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 (mathematics)
In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is an algebraic structure similar to a group but without requiring inverse or closure properties.
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Category of groups
In mathematics, the category Grp has the class of all groups for objects and group homomorphisms for morphisms.
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Category of magmas
In mathematics, the category of magmas, denoted Mag, has as objects sets with a binary operation, and morphisms given by homomorphisms of operations (in the universal algebra sense).
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Category of sets
In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are 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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Character (computing)
In computer and machine-based telecommunications terminology, a character is a unit of information that roughly corresponds to a grapheme, grapheme-like unit, or symbol, such as in an alphabet or syllabary in the written form of a natural language.
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Class (set theory)
In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share.
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Closed manifold
In mathematics, a closed manifold is a type of topological space, namely a compact manifold without boundary.
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Closure (mathematics)
A set has closure under an operation if performance of that operation on members of the set always produces a member of the same set; in this case we also say that the set is closed under the operation.
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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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Complex number
A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.
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Computer science
Computer science deals with the theoretical foundations of information and computation, together with practical techniques for the implementation and application of these foundations.
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Concatenation
In formal language theory and computer programming, string concatenation is the operation of joining character strings end-to-end.
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Concurrent computing
Concurrent computing is a form of computing in which several computations are executed during overlapping time periods—concurrently—instead of sequentially (one completing before the next starts).
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Connected sum
In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds.
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Constant (mathematics)
In mathematics, the adjective constant means non-varying.
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Constant function
In mathematics, a constant function is a function whose (output) value is the same for every input value.
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Cyclic group
In algebra, a cyclic group or monogenous group is a group that is generated by a single element.
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Data structure
In computer science, a data structure is a data organization and storage format that enables efficient access and modification.
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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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Empty string
In formal language theory, the empty string, or empty word is the unique string of length zero.
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Endomorphism
In mathematics, an endomorphism is a morphism (or homomorphism) from a mathematical object to itself.
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Equivalence of categories
In category theory, an abstract branch of mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same".
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Exclusive or
Exclusive or or exclusive disjunction is a logical operation that outputs true only when inputs differ (one is true, the other is false).
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Finite-state machine
A finite-state machine (FSM) or finite-state automaton (FSA, plural: automata), finite automaton, or simply a state machine, is a mathematical model of computation.
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Fold (higher-order function)
In functional programming, fold (also termed reduce, accumulate, aggregate, compress, or inject) refers to a family of higher-order functions that analyze a recursive data structure and through use of a given combining operation, recombine the results of recursively processing its constituent parts, building up a return value.
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Free monoid
In abstract algebra, the free monoid on a set is the monoid whose elements are all the finite sequences (or strings) of zero or more elements from that set, with string concatenation as the monoid operation and with the unique sequence of zero elements, often called the empty string and denoted by ε or λ, as the identity element.
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Free object
In mathematics, the idea of a free object is one of the basic concepts of abstract algebra.
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Function composition
In mathematics, function composition is the pointwise application of one function to the result of another to produce a third function.
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Functor
In mathematics, a functor is a map between categories.
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Green's relations
In mathematics, Green's relations are five equivalence relations that characterise the elements of a semigroup in terms of the principal ideals they generate.
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Grothendieck group
In mathematics, the Grothendieck group construction in abstract algebra constructs an abelian group from a commutative monoid M in the most universal way in the sense that any abelian group containing a homomorphic image of M will also contain a homomorphic image of the Grothendieck group of M. The Grothendieck group construction takes its name from the more general construction in category theory, introduced by Alexander Grothendieck in his fundamental work of the mid-1950s that resulted in the development of K-theory, which led to his proof of the Grothendieck–Riemann–Roch theorem.
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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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Group action
In mathematics, an action of a group is a formal way of interpreting the manner in which the elements of the group correspond to transformations of some space in a way that preserves the structure of that space.
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Group homomorphism
In mathematics, given two groups, (G, ∗) and (H, ·), a group homomorphism from (G, ∗) to (H, ·) is a function h: G → H such that for all u and v in G it holds that where the group operation on the left hand side of the equation is that of G and on the right hand side that of H. From this property, one can deduce that h maps the identity element eG of G to the identity element eH of H, and it also maps inverses to inverses in the sense that Hence one can say that h "is compatible with the group structure".
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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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History monoid
In mathematics and computer science, a history monoid is a way of representing the histories of concurrently running computer processes as a collection of strings, each string representing the individual history of a process.
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Homeomorphism
In the mathematical field of topology, a homeomorphism or topological isomorphism or bi continuous function is a continuous function between topological spaces that has a continuous inverse function.
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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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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 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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Identity function
Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation or identity map or identity transformation, is a function that always returns the same value that was used as its argument.
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Index set
In mathematics, an index set is a set whose members label (or index) members of another set.
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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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Integer
An integer (from the Latin ''integer'' meaning "whole")Integer 's first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").
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Inverse element
In abstract algebra, the idea of an inverse element generalises concepts of a negation (sign reversal) in relation to addition, and a reciprocal in relation to multiplication.
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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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Kleene algebra
In mathematics, a Kleene algebra (named after Stephen Cole Kleene) is an idempotent (and thus partially ordered) semiring endowed with a closure operator.
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Krohn–Rhodes theory
In mathematics and computer science, the Krohn–Rhodes theory (or algebraic automata theory) is an approach to the study of finite semigroups and automata that seeks to decompose them in terms of elementary components.
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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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Lexicographical order
In mathematics, the lexicographic or lexicographical order (also known as lexical order, dictionary order, alphabetical order or lexicographic(al) product) is a generalization of the way words are alphabetically ordered based on the alphabetical order of their component letters.
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Logical biconditional
In logic and mathematics, the logical biconditional (sometimes known as the material biconditional) is the logical connective of two statements asserting "P if and only if Q", where P is an antecedent and Q is a consequent.
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Logical conjunction
In logic, mathematics and linguistics, And (∧) is the truth-functional operator of logical conjunction; the and of a set of operands is true if and only if all of its operands are true.
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Logical disjunction
In logic and mathematics, or is the truth-functional operator of (inclusive) disjunction, also known as alternation; the or of a set of operands is true if and only if one or more of its operands is true.
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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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MapReduce
MapReduce is a programming model and an associated implementation for processing and generating big data sets with a parallel, distributed algorithm on a cluster.
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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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Matrix (mathematics)
In mathematics, a matrix (plural: matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.
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Matrix addition
In mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together.
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Matrix multiplication
In mathematics, matrix multiplication or matrix product is a binary operation that produces a matrix from two matrices with entries in a field, or, more generally, in a ring or even a semiring.
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Monad (functional programming)
In functional programming, a monad is a design pattern that defines how functions, actions, inputs, and outputs can be used together to build generic types, with the following organization.
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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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Monoid (category theory)
In category theory, a monoid (or monoid object) (M, μ, η) in a monoidal category (C, ⊗, I) is an object M together with two morphisms.
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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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Multiset
In mathematics, a multiset (aka bag or mset) is a modification of the concept of a set that, unlike a set, allows for multiple instances for each of its elements.
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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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Null semigroup
In mathematics, a null semigroup (also called a zero semigroup) is a semigroup with an absorbing element, called zero, in which the product of any two elements is zero.
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Numerical semigroup
In mathematics, a numerical semigroup is a special kind of a semigroup.
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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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Oxford University Press
Oxford University Press (OUP) is the largest university press in the world, and the second oldest after Cambridge University Press.
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Parallel computing
Parallel computing is a type of computation in which many calculations or the execution of processes are carried out concurrently.
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Partially ordered group
In abstract algebra, a partially ordered group is a group (G,+) equipped with a partial order "≤" that is translation-invariant; in other words, "≤" has the property that, for all a, b, and g in G, if a ≤ b then a+g ≤ b+g and g+a ≤ g+b.
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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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Plactic monoid
In mathematics, the plactic monoid is the monoid of all words in the alphabet of positive integers modulo Knuth equivalence.
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Pointwise
In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some function f. An important class of pointwise concepts are the pointwise operations — operations defined on functions by applying the operations to function values separately for each point in the domain of definition.
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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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Prefix sum
In computer science, the prefix sum, cumulative sum, inclusive scan, or simply scan of a sequence of numbers is a second sequence of numbers, the sums of prefixes (running totals) of the input sequence: For instance, the prefix sums of the natural numbers are the triangular numbers: |- !input numbers | 1 || 2 || 3 || 4 || 5 || 6 ||...
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Preorder
In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive.
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Presentation of a group
In mathematics, one method of defining a group is by a presentation.
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Process calculus
In computer science, the process calculi (or process algebras) are a diverse family of related approaches for formally modelling concurrent systems.
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Product of group subsets
In mathematics, one can define a product of group subsets in a natural way.
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Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator.
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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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Semiautomaton
In mathematics and theoretical computer science, a semiautomaton is a deterministic finite automaton having inputs but no output.
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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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Semigroup action
In algebra and theoretical computer science, an action or act of a semigroup on a set is a rule which associates to each element of the semigroup a transformation of the set in such a way that the product of two elements of the semigroup (using the semigroup operation) is associated with the composite of the two corresponding transformations.
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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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Semiring
In abstract algebra, a semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse.
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Sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed.
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Set (mathematics)
In mathematics, a set is a collection of distinct objects, considered as an object in its own right.
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Singleton (mathematics)
In mathematics, a singleton, also known as a unit set, is a set with exactly one element.
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Springer Science+Business Media
Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
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Star height problem
The star height problem in formal language theory is the question whether all regular languages can be expressed using regular expressions of limited star height, i.e. with a limited nesting depth of Kleene stars.
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String (computer science)
In computer programming, a string is traditionally a sequence of characters, either as a literal constant or as some kind of variable.
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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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Syntactic monoid
In mathematics and computer science, the syntactic monoid M(L) of a formal language L is the smallest monoid that recognizes the language L.
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Torus
In geometry, a torus (plural tori) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis coplanar with the circle.
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Trace monoid
In computer science, a trace is a set of strings, wherein certain letters in the string are allowed to commute, but others are not.
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Transformation semigroup
In algebra, a transformation semigroup (or composition semigroup) is a collection of functions from a set to itself that is closed under function composition.
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Transition system
In theoretical computer science, a transition system is a concept used in the study of computation.
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Tree traversal
In computer science, tree traversal (also known as tree search) is a form of graph traversal and refers to the process of visiting (checking and/or updating) each node in a tree data structure, exactly once.
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Trivial group
In mathematics, a trivial group is a group consisting of a single element.
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Tuple
In mathematics, a tuple is a finite ordered list (sequence) of elements.
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Vedic square
In Indian mathematics, a Vedic square is a variation on a typical 9 × 9 multiplication table where the entry in each cell is the digital root of the product of the column and row headings i.e. the remainder when the product of the row and column headings is divided by 9 (with remainder 0 represented by 9).
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Zerosumfree monoid
In abstract algebra, an additive monoid (M, 0, +) is said to be zerosumfree, conical, centerless or positive if nonzero elements do not sum to zero.
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0
0 (zero) is both a number and the numerical digit used to represent that number in numerals.
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1
1 (one, also called unit, unity, and (multiplicative) identity) is a number, numeral, and glyph.
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Abelian monoid, Commutative monoid, Complete monoid, Continuous monoid, Finitely generated monoid, Monoid (algebra), Monoid homomorphism, Monoid morphism, Monoid presentation, Monoids, Submonoid.
References
[1] https://en.wikipedia.org/wiki/Monoid
