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Special classes of semigroups

Index Special classes of semigroups

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

74 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, Empty semigroup, Empty set, Epigroup, Exponential function, 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, Nilsemigroup, Nowhere commutative semigroup, Null semigroup, Numerical semigroup, Ordered semigroup, Orthodox semigroup, Oxford University Press, Partial function, Partially ordered set, Presentation of a monoid, Property (philosophy), ..., Recognizable set, Regular semigroup, Semigroup, Semigroup Forum, Semigroup with involution, Semilattice, Set (mathematics), Springer Science+Business Media, Structure, Subgroup, Subset, Symmetric inverse semigroup, Syntactic monoid, Theory, Thiruvananthapuram, Topological semigroup, Topology, Transformation semigroup, Trivial semigroup, University of Kerala, Variety (universal algebra), Variety of finite semigroups, Vector space, World Scientific. Expand index (24 more) »

Algebra

Algebra (from Arabic "al-jabr", literally 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, 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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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 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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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 the mathematics of binary relations, the composition relations is a concept of forming a new relation from two given relations R and S. The composition of relations is called relative multiplication in the calculus of relations.

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CRC Press

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

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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 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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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 function

In mathematics, an exponential function is a function of the form in which the argument occurs as an exponent.

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Field (mathematics)

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.

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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 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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India

India (IAST), also called the Republic of India (IAST), is a country in South Asia.

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Inverse semigroup

In group theory, 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.

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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, sometimes shortened to map, refers to either a function, often with some sort of special structure, or a morphism in category theory, which generalizes the idea of a function.

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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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Monogenic semigroup

In mathematics, a monogenic semigroup is a semigroup generated by 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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Nilsemigroup

In mathematics, and more precisely in semigroup theory, a a nilsemigroup or a nilpotent semigroup is a semigroup whose every elements are nilpotent.

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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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Ordered semigroup

In mathematics, an ordered semigroup is a semigroup (S,•) together with a partial order ≤ that is compatible with the semigroup operation, meaning that x ≤ y implies z•x ≤ z•y and x•z ≤ y•z for all x, y, z in S. If S is a group and it is ordered as a semigroup, one obtains the notion of ordered group, and similarly if S is a monoid it may be called ordered monoid.

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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 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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Presentation of a monoid

In algebra, a presentation of a monoid (respectively, a presentation of a semigroup) is a description of a monoid (respectively, a semigroup) in terms of a set of generators and a set of relations on the free monoid (respectively, the free semigroup) generated by.

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Property (philosophy)

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

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Recognizable set

In computer science, more precisely in automata theory, a recognizable set of a monoid is a subset that can be distinguished by some morphism to a finite monoid.

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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 Forum

Semigroup Forum (print, electronic) is a mathematics research journal published by Springer.

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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, 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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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 group theory, a branch of 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, 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 (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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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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Theory

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

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Thiruvananthapuram

Thiruvananthapuram, also known as Trivandrum, is the capital and the largest city of the Indian state of Kerala.

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Topological semigroup

In mathematics, a topological semigroup is a semigroup which is simultaneously a topological space, and whose semigroup operation is continuous.

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Topology

In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing.

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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

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

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Variety (universal algebra)

In the mathematical subject of universal algebra, a variety of algebras is the class of all algebraic structures of a given signature satisfying a given set of identities.

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Variety of finite semigroups

In mathematics, and more precisely in semigroup theory, a variety of finite semigroups is a set of semigroups having some nice algebraic properties.

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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.

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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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