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Special classes of semigroups

In mathematics, a semigroup is a nonempty set together with an associative binary operation. [1]

66 relations: Algebra, Algebraic structure, American Mathematical Society, Aperiodic semigroup, Archimedean property, Associative property, Band (mathematics), Bicyclic semigroup, Bijection, Binary operation, Binary relation, Brandt semigroup, Cancellative semigroup, Cardinality, Class (set theory), Classification theorem, Clifford semigroup, Commutative property, Completely regular semigroup, Composition of relations, CRC Press, E-dense semigroup, Empty semigroup, Empty set, Epigroup, Exponential, Field (mathematics), Finite set, Free monoid, Function composition, Group (mathematics), India, Inverse semigroup, John Wiley & Sons, Linear map, Map (mathematics), Mathematics, Matrix (mathematics), Monogenic semigroup, Monoid, Nowhere commutative semigroup, Null semigroup, Numerical semigroup, Orthodox semigroup, Oxford University Press, Partial function, Partially ordered set, Property (philosophy), Regular semigroup, Semigroup, ..., Semigroup with involution, Semilattice, Set (mathematics), Springer Science+Business Media, Structure, Subgroup, Subset, Symmetric inverse semigroup, Theory, Thiruvananthapuram, Topology, Transformation semigroup, Trivial semigroup, University of Kerala, Vector space, World Scientific. Expand index (16 more) »

Algebra

Algebra (from Arabic and Farsi "al-jabr" meaning "reunion of broken parts") is one of the broad parts of mathematics, together with number theory, geometry and analysis.

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Algebraic structure

In mathematics, and more specifically in abstract algebra, the term algebraic structure generally refers to a set (called carrier set or underlying set) with one or more finitary operations defined on it that satisfies a some list of axioms.

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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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Aperiodic semigroup

In mathematics, an aperiodic semigroup is a semigroup S such that every element x ∈ S is aperiodic, that is, for each x there exists a positive integer n such that xn.

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Archimedean property

In abstract algebra and analysis, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some algebraic structures, such as ordered or normed groups, and fields.

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Associative property

In mathematics, the associative property is a property of some binary operations.

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Band (mathematics)

In mathematics, a band (also called idempotent semigroup) is a semigroup in which every element is idempotent (in other words equal to its own square).

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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 every element of one set is paired with exactly one element of the other set, and every 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 (more formally, an operation whose arity is two, and whose two domains and one codomain are (subsets of) the same set).

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Binary relation

In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2.

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Brandt semigroup

In mathematics, Brandt semigroups are completely 0-simple inverse semigroups.

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Cancellative semigroup

In mathematics, a cancellative semigroup (also called a cancellation semigroup) is a semigroup having the cancellation property.

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Cardinality

In mathematics, the cardinality of a set is a measure of the "number of elements of the set".

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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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Classification theorem

In mathematics, a classification theorem answers the classification problem "What are the objects of a given type, up to some equivalence?".

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Clifford semigroup

A Clifford semigroup (sometimes also called "inverse Clifford semigroup") is a completely regular inverse semigroup.

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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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Completely regular semigroup

In mathematics, a completely regular semigroup is a semigroup in which every element is in some subgroup of the semigroup.

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Composition of relations

In mathematics, the composition of binary relations is a concept of forming a new relation from two given relations R and S, having as its most well-known special case the composition of functions.

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CRC Press

The CRC Press, LLC is a publishing group that specializes in producing technical books.

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E-dense semigroup

In abstract algebra, an E-dense semigroup (also called an E-inversive semigroup) is a semigroup in which every element a has at least one weak inverse x, meaning that xax.

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Empty semigroup

In mathematics, a semigroup with no elements (the empty semigroup) is a semigroup in which the underlying set is the empty set.

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Empty set

In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero.

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Epigroup

In abstract algebra, an epigroup is a semigroup in which every element has a power that belongs to a subgroup.

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Exponential

Exponential may refer to any of several mathematical topics related to exponentiation, including.

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Field (mathematics)

In abstract algebra, a field is a nonzero commutative division ring, or equivalently a ring whose nonzero elements form an abelian group under multiplication.

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Finite set

In mathematics, a finite set is a set that has a finite number of elements.

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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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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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Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set of elements together with an operation that combines any two elements to form a third element.

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India

India, officially the Republic of India, is a country in South Asia.

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Inverse semigroup

In mathematics, an inverse semigroup (occasionally called an inversion semigroup) S is a semigroup in which every element x in S has a unique inverse y in S in the sense that x.

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John Wiley & Sons

John Wiley & Sons, Inc., also referred to as Wiley, is a global publishing company that specializes in academic publishing and markets its products to professionals and consumers, students and instructors in higher education, and researchers and practitioners in scientific, technical, medical, and scholarly fields.

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Linear map

In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication.

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Map (mathematics)

In mathematics, the term mapping, usually shortened to map, refers to either.

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Mathematics

Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), 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—that is interpreted and manipulated in certain prescribed ways.

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Monogenic semigroup

In mathematics, a monogenic semigroup is a semigroup generated by a set containing only a single element.

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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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Nowhere commutative semigroup

In mathematics, a nowhere commutative semigroup is a semigroup S such that, for all a and b in S, if ab.

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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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Orthodox semigroup

In mathematics, an orthodox semigroup is a regular semigroup whose set of idempotents forms a subsemigroup.

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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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Partial function

In mathematics, a partial function from X to Y (written as) 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 (or poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set.

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Property (philosophy)

In modern philosophy and mathematics, a property is a characteristic of an object; a red object is said to have the property of redness.

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Regular semigroup

In mathematics, a regular semigroup is a semigroup S in which every element is regular, i.e., for each element a, there exists an element x such that axa.

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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 with involution

In mathematics, particularly in abstract algebra, a semigroup with involution or a *-semigroup is a semigroup equipped with an involutive anti-automorphism, which—roughly speaking—brings it closer to a group because this involution, considered as unary operator, exhibits certain fundamental properties of the operation of taking the inverse in a group: uniqueness, double application "cancelling itself out", and the same interaction law with the binary operation as in the case of the group inverse.

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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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Set (mathematics)

In mathematics, a set is a collection of distinct objects, considered as an object in its own right.

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Springer Science+Business Media

Springer Science+Business Media or Springer is a global publishing company that publishes books, e-books and peer-reviewed journals in science, technical and medical (STM) publishing.

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Structure

Structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized.

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Subgroup

In mathematics, given a group G under a binary operation ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗.

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Subset

In mathematics, especially in set theory, 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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Symmetric inverse semigroup

In abstract algebra, the set of all partial bijections on a set X (aka one-to-one partial transformations) forms an inverse semigroup, called the symmetric inverse semigroup (actually a monoid) on X. The conventional notation for the symmetric inverse semigroup on a set X is \mathcal_X or \mathcal_X In general \mathcal_X is not commutative.

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Theory

Theory is a contemplative and rational type of abstract or generalizing thinking, or the results of such thinking.

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Thiruvananthapuram

Thiruvananthapuram (Malayalam Tiruvaṉantapuram), also known as Trivandrum, is the capital city of the Indian state of Kerala.

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Topology

In mathematics, topology (from the Greek τόπος, place, and λόγος, study), is the study of topological spaces.

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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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Trivial semigroup

In mathematics, a trivial semigroup (a semigroup with one element) is a semigroup for which the cardinality of the underlying set is one.

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University of Kerala

The University of Kerala (UoK), formerly the University of Travancore, is an affiliating university located in Trivandrum in the south Indian state of Kerala, India.

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Vector space

A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars in this context.

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World Scientific

World Scientific Publishing is an academic publisher of scientific, technical, and medical books and journals headquartered in Singapore.

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0-simple, 0-simple semigroup, Commutative semigroup, Completely 0-simple semigroup, Completely 0-simple semigroups, Completely simple semigroup, Eventually regular semigroup, Pseudo-inverse semigroup, Pseudoinverse semigroup, Simple semigroup.

References

[1] https://en.wikipedia.org/wiki/Special_classes_of_semigroups

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