Global Information Lookup Global Information

Semigroup with involution information


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".
  • The same interaction law with the binary operation as in the case of the group inverse.

It is thus not a surprise that any group is a semigroup with involution. However, there are significant natural examples of semigroups with involution that are not groups.

An example from linear algebra is the multiplicative monoid of real square matrices of order n (called the full linear monoid). The map which sends a matrix to its transpose is an involution because the transpose is well defined for any matrix and obeys the law (AB)T = BTAT, which has the same form of interaction with multiplication as taking inverses has in the general linear group (which is a subgroup of the full linear monoid). However, for an arbitrary matrix, AAT does not equal the identity element (namely the diagonal matrix). Another example, coming from formal language theory, is the free semigroup generated by a nonempty set (an alphabet), with string concatenation as the binary operation, and the involution being the map which reverses the linear order of the letters in a string. A third example, from basic set theory, is the set of all binary relations between a set and itself, with the involution being the converse relation, and the multiplication given by the usual composition of relations.

Semigroups with involution appeared explicitly named in a 1953 paper of Viktor Wagner (in Russian) as result of his attempt to bridge the theory of semigroups with that of semiheaps.[1]

  1. ^ Christopher Hollings (2014). Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups. American Mathematical Society. p. 265. ISBN 978-1-4704-1493-1.

and 23 Related for: Semigroup with involution information

Request time (Page generated in 0.9065 seconds.)

Semigroup with involution

Last Update:

mathematics, particularly in abstract algebra, a semigroup with involution or a *-semigroup is a semigroup equipped with an involutive anti-automorphism, which—roughly...

Word Count : 3600

Semigroup

Last Update:

we mention: regular semigroups, orthodox semigroups, semigroups with involution, inverse semigroups and cancellative semigroups. There are also interesting...

Word Count : 4675

Inverse element

Last Update:

an involution, and typically denoted by a* Clearly a group is both an I-semigroup and a *-semigroup. A class of semigroups important in semigroup theory...

Word Count : 4478

Antihomomorphism

Last Update:

composition of an antihomomorphism with a homomorphism gives another antihomomorphism. Semigroup with involution Jacobson, Nathan (1943). The Theory...

Word Count : 644

Converse relation

Last Update:

relation to the converse relation is an involution, so it induces the structure of a semigroup with involution on the binary relations on a set, or, more...

Word Count : 1725

Inverse semigroup

Last Update:

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...

Word Count : 3748

Distributive property

Last Update:

} which is taken as an axiom in the more general context of a semigroup with involution, has sometimes been called an antidistributive property (of inversion...

Word Count : 2998

Special classes of semigroups

Last Update:

mathematics, a semigroup is a nonempty set together with an associative binary operation. A special class of semigroups is a class of semigroups satisfying...

Word Count : 428

Binary relation

Last Update:

operation on B ( X ) {\displaystyle {\mathcal {B}}(X)} , it forms a semigroup with involution. Some important properties that a homogeneous relation R {\displaystyle...

Word Count : 8911

Presentation of a monoid

Last Update:

presentation of a monoid (or a presentation of a semigroup) is a description of a monoid (or a semigroup) in terms of a set Σ of generators and a set of...

Word Count : 785

Composition of relations

Last Update:

This property makes the set of all binary relations on a set a semigroup with involution. The composition of (partial) functions (that is, functional relations)...

Word Count : 2834

Partial isometry

Last Update:

can be defined in the more abstract setting of a semigroup with involution; the definition coincides with the one herein. In finite-dimensional vector spaces...

Word Count : 1275

Additive inverse

Last Update:

Monoid Inverse function Involution (mathematics) Multiplicative inverse Reflection (mathematics) Reflection symmetry Semigroup Tussy, Alan; Gustafson,...

Word Count : 887

Outline of algebraic structures

Last Update:

single binary operation over S. Semigroup: an associative magma. Monoid: a semigroup with identity element. Group: a monoid with a unary operation (inverse)...

Word Count : 2214

Baer ring

Last Update:

operators on a Hilbert space are a Baer ring and is also a Baer *-ring with the involution * given by the adjoint. von Neumann algebras are examples of all...

Word Count : 793

Quantale

Last Update:

sometimes referred to as complete residuated semigroups. A quantale is a complete lattice Q {\displaystyle Q} with an associative binary operation ∗ : Q ×...

Word Count : 554

Complemented lattice

Last Update:

complemented lattice. An orthocomplementation on a complemented lattice is an involution that is order-reversing and maps each element to a complement. An orthocomplemented...

Word Count : 876

Algebra over a field

Last Update:

underlying Banach space, which turns them into Banach algebras. If an involution is given as well, we obtain B*-algebras and C*-algebras. These are studied...

Word Count : 2913

Composition algebra

Last Update:

N(xy)=N(x)N(y)} for all x and y in A. A composition algebra includes an involution called a conjugation: x ↦ x ∗ . {\displaystyle x\mapsto x^{*}.} The quadratic...

Word Count : 1319

Symmetric group

Last Update:

Sn is generated by involutions (2-cycles, which have order 2), so the only non-trivial maps Sn → Cp are to S2 and all involutions are conjugate, hence...

Word Count : 6130

Relation algebra

Last Update:

algebra, a relation algebra is a residuated Boolean algebra expanded with an involution called converse, a unary operation. The motivating example of a relation...

Word Count : 2546

Group action

Last Update:

See semigroup action. Instead of actions on sets, we can define actions of groups and monoids on objects of an arbitrary category: start with an object...

Word Count : 5591

Ternary relation

Last Update:

ISBN 3-540-63246-8 Novák, Vítězslav (1996), "Ternary structures and partial semigroups", Czechoslovak Mathematical Journal, 46 (1): 111–120, hdl:10338.dmlcz/127275...

Word Count : 735

PDF Search Engine © AllGlobal.net