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In mathematics, a semigroup is an algebraic structure consisting of a set S closed under an associative binary operation. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Algebra, a branch of Pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, In Mathematics, associativity is a property that a Binary operation can have In Mathematics, a binary operation is a calculation involving two Operands, in other words an operation whose Arity is two In other words, a semigroup is an associative magma. In Abstract algebra, a magma (or groupoid) is a basic kind of Algebraic structure. The terminology is derived from the anterior notion of a group. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element

The operation of a semigroup is most often denoted multiplicatively, that is, x\cdot y or simply xy denotes the result of applying the semigroup operation to the ordered pair (x, y).

The formal study of semigroups began in the early 20th century. The twentieth century of the Common Era began on Since the 1950s, the theory of finite semigroups has been of particular importance in theoretical computer science because of the natural link between finite semigroups and finite automata. The 1950s Decade refers to the years of 1950 to 1959 inclusive Theoretical computer science is the collection of topics of Computer science that focuses on the more abstract logical and mathematical aspects of Computing, such

Contents

Formal definition

A semigroup formally consists of a pair (S,\cdot_S) of a set S and a (total) binary function \cdot_S: S \times S \rightarrow S called the operation of the semigroup. For convenience, the application of the function \cdot_S to the pair (x,y) is simply denoted as x \cdot_S y or x \cdot y. The operation is required to be associative, i. e. to satisfy (x \cdot y) \cdot z = x \cdot (y \cdot z) for any x,y,z \in S. As is common practice in abstract algebra, one usually refers to the pair (S,\cdot_S) as S when the operation used is clear from the context.

Some authors require semigroups to be non-empty. Others use the term semigroup synonymously with monoid, that is, they assume that a semigroup has an identity element. In Abstract algebra, a branch of Mathematics, a monoid is an Algebraic structure with a single Associative Binary operation 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 In the remainder of this article, the term semigroup will be used in the widest sense, that is, a semigroup may be empty, and even if non-empty it need not include an identity element.

As noted above, a monoid is a semigroup with an identity element.

Embeddings

Any semigroup S may be embedded into a monoid (generally denoted as S1) simply by adjoining an element e not in S and defining es = s = se for all s ∈ S ∪ {e}.

A commutative semigroup can be embedded into a group if and only if it has the cancellation property. ↔ In Mathematics, the notion of cancellative is a generalization of the notion of Invertible.

Examples of semigroups

Structure of semigroups

This section sets out concepts useful for understanding the structure of semigroups. Two semigroups S and T are said to be isomorphic if there is a bijection f : S ↔ T with the property that, for any elements a, b in S, f(ab) = f(a)f(b). In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property In this case, T and S are also isomorphic, and for the purposes of semigroup theory, the two semigroups are identical.

If A and B are subsets of some semigroup, then AB denotes the set { ab | a in A and b in B }. A subset A of a semigroup S is called a subsemigroup if it is closed under the semigroup operation, that is, AA is a subset of A. Let A be nonempty. A is called a right ideal if AS is a subset of A, and a left ideal if SA is a subset of A. If A is both a left ideal and a right ideal then it is called an ideal (or a two-sided ideal). The intersection of two ideals is also an ideal, so a semigroup can have at most one minimal ideal. An example of semigroup with no minimal ideal is the set of positive integers under addition. The minimal ideal of a commutative semigroup, when it exists, is a group. In Mathematics, commutativity is the ability to change the order of something without changing the end result

Green's relations are important tools for analysing the ideals of a semigroup, and related notions of structure. In Mathematics, Green's relations are five Equivalence relations that characterise the elements of a Semigroup in terms of the Principal ideals

If S is a semigroup, then the intersection of any collection of subsemigroups of S is also a subsemigroup of S. So the subsemigroups of S form a 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 For any subset A of S there is a smallest subsemigroup T of S which contains A, and we say that A generates T. A single element x of S generates the subsemigroup { xn | n is a positive integer }. If this is finite, then x is said to be of finite order, otherwise it is of infinite order. A semigroup is said to be periodic if all of its elements are of finite order. A semigroup generated by a single element is said to be monogenic (or cyclic). If a monogenic semigroup is infinite then it is isomorphic to the semigroup of positive integers with the operation of addition. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French If it is finite and nonempty, then it must contain at least one idempotent. Idempotence ˌaɪdɨmˈpoʊtəns describes the property of operations in Mathematics and Computer science which means that multiple applications of the operation It follows that every nonempty periodic semigroup has at least one idempotent.

A subsemigroup which is also a group is called a subgroup. In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of There is a close relationship between the subgroups of a semigroup and its idempotents. Each subgroup contains exactly one idempotent, namely the identity element of the subgroup. For each idempotent e of the semigroup there is a unique maximal subgroup containing e. Each maximal subgroup arises in this way, so there is a one-to-one correspondence between idempotents and maximal subgroups. Here the term maximal subgroup differs from its standard use in group theory. In Mathematics, the term maximal subgroup is used to mean slightly different things in different areas of Algebra.

More can often be said when the order is finite. For example, every nonempty finite semigroup is periodic, and has a minimal ideal and at least one idempotent. iDEAL is an Internet payment method in The Netherlands, based on online banking Idempotence ˌaɪdɨmˈpoʊtəns describes the property of operations in Mathematics and Computer science which means that multiple applications of the operation For more on the structure of finite semigroups, see Krohn-Rhodes theory.

Semigroup methods in partial differential equations

Semigroup theory can be used to study some problems in the field of partial differential equations. In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i Roughly speaking, the semigroup approach is to regard a time-dependent partial differential equation as an ordinary differential equation on a function space. In Mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one Independent variable, and one or more of its For example, consider the following initial/boundary value problem for the heat equation on the spatial interval (0, 1) ⊂ R and times t ≥ 0:

\begin{cases} \partial_{t} u(t, x) = \partial_{x}^{2} u(t, x), & x \in (0, 1), t > 0; \\ u(t, x) = 0, & x \in \{ 0, 1 \}, t > 0; \\ u(t, x) = u_{0} (x), & x \in (0, 1), t = 0. \end{cases}

Let X be the Lp space L2((0, 1); R) and let A be the second-derivative operator with domain

D(A) = \big\{ u \in H^{2} ((0, 1); \mathbf{R}) \big| u(0) = u(1) = 0 \big\}.

Then the above initial/boundary value problem can be interpreted as an initial value problem for an ordinary differential equation on the space X:

\begin{cases} \dot{u}(t) = A u (t); \\ u(0) = u_{0}. \end{cases}

On an heuristic level, the solution to this problem "ought" to be u(t) = exp(tA)u0. The heat equation is an important Partial differential equation which describes the distribution of Heat (or variation in temperature in a given region over time In Mathematics, an interval is a set of Real numbers with the property that any number that lies between two numbers in the set is also included in the set In Mathematics, the Lp and ℓp spaces are spaces of p-power integrable functions, and corresponding In Mathematics, the domain of a given function is the set of " Input " values for which the function is defined However, for a rigorous treatment, a meaning must be given to the exponential of tA. As a function of t, exp(tA) is a semigroup of operators from X to itself, taking the initial state u0 at time t = 0 to the state u(t) = exp(tA)u0 at time t. The operator A is said to be the infinitesimal generator of the semigroup. In Mathematics, a C 0-semigroup, also known as a (strongly continuous one-parameter semigroup, is a continuous morphism from ( R ++

History

The formal study of semigroups came somewhat later than that of other algebraic structures such as groups or rings in the mid 19th century. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real The 19th century of the Common Era began on January 1, 1801 and ended on December 31, 1900, according to the Gregorian calendar A number of sources[1][2] attribute the first use of the term (in French) to J. -A. de Séguier in Élements de la Théorie des Groupes Abstraits (Elements of the Theory of Abstract Groups) in 1904. The term is used in English in 1908 in Harold Hinton's Theory of Groups of Finite Order. In 1970, a new periodical called Semigroup Forum (currently edited by Springer Verlag) became one of the rare mathematical journals devoted entirely to semigroup theory. Springer Science+Business Media or Springer (ˈʃpʁɪŋɐ is a worldwide Publishing company based in Germany, which publishes textbooks academic

Anton Suschkewitsch is often credited with obtaining the first non-trivial results about semigroups. His 1928 paper Über die endlichen Gruppen ohne das Gesetz der eindeutigen Umkehrbarkeit (On finite groups without the rule of unique invertibility) determined the structure of finite simple semigroups and showed that the minimal ideal (or Green's relations J-class) of a finite semigroup is simple[2]. In Mathematics, Green's relations are five Equivalence relations that characterise the elements of a Semigroup in terms of the Principal ideals From that point on, the foundations of semigroup theory were further laid by David Rees, James Alexander Green, Evgenii Sergeevich Lyapin, Alfred H. David Rees MA, ScD Cantab, FIMA, FRS (Mathematics is an emeritus professor of pure mathematics at the University of Exeter James Alexander (Sandy Green (b1926 is a mathematician and retired Professor at the Mathematics Institute at the University of Warwick, who is still active in the Clifford and Gordon Preston. The latter two published a monograph on semigroup theory in 1961.

The theory of finite semigroups is arguably more developed than its infinite counterpart. This stems particularly from the notion of syntactic semigroup and the ensuing links between pseudo-varieties of semigroups and so-called varieties of formal languages which have proved particularly fruitful in finite automata theory[3]. In Mathematics, the syntactic monoid M ( L) of a Formal language L is the smallest Monoid that recognizes the

See also

References

Notes

  1. ^ http://members.aol.com/jeff570/s.html Earliest Known Uses of Some of the Words of Mathematics
  2. ^ a b http://www-users.york.ac.uk/~cdh500/suschkewitsch3.pdf An account of Suschkewitsch's paper by Christopher Hollings
  3. ^ Varieties of Formal Languages, J. É. Pin, Plenum Press, 1986.

Dictionary

semigroup

-noun

  1. (mathematics) Any set for which there is a binary operation that is both closed and associative
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