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semigroup

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sem·i·group

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(sĕm'ē-grūp', sĕm'ī-) n. MathematicsA set for which there is a  binary operation that is closed and associative.

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Semigroup

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A semigroup is an algebraic  structure consisting of a set S together with an  associative binary  operation. In other words, a semigroup is an associative magma. The  terminology is derived from the anterior notion of a group. A semigroup  differs from a group in that for each of its elements there might not exist an  inverse; further, there might not exist an identity element.
The binary operation of a semigroup is most often denoted  multiplicatively: ,  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. Semigroups are important in many areas of mathematics because they are  the abstract algebraic underpinning of "memoryless" systems: time-dependent  systems that start from scratch at each iteration. In applied mathematics,  semigroups are fundamental models for linear  time-invariant systems. In partial differential equations, a semigroup is associated to any  equation whose spatial evolution is independent of time. The theory of finite  semigroups has been of particular importance in theoretical computer science since the 1950s because of the  natural link between finite semigroups and finite  automata. In probability  theory, semigroups are associated with Markov  processes (Feller  1971).

Contents [hide]

1 Definition
2 Examples of semigroups
3 Basic concepts

3.1 Identity and zero
3.2 Subsemigroups and  ideals
3.3 Homomorphisms and  congruences

4 Structure of semigroups
5 Special classes of  semigroups
6 Group of fractions
7 Semigroup methods in partial  differential equations
8 History
9 Generalizations
11 Notes
12 References

Definition
A semigroup is a set, S, together with a  binary  operation ""  that satisfies:
Closure For all a, b in S, the result of the operation a · b is also in S. Associativity For all a, b and c in S, the equation (a · b) · c =  a · (b · c)  holds.
And in mathematical notation we have:
and .
More compactly, a semigroup is an associative magma.
Examples of semigroups

Empty  semigroup: the empty set forms a semigroup with the empty function as the  binary operation.
Semigroup with one element: there is essentially just one, the  singleton {a} with operation a ·  a = a.
Semigroup with two elements: there are five which are  essentially different.
The set of positive integers with  addition.
Square nonnegative  matrices with matrix multiplication.
Any ideal of a  ring with the  multiplication of the ring.
The set of all finite strings over a fixed alphabet Σ with concatenation of strings as  the semigroup operation — the so-called "free  semigroup over Σ". With the empty string included, this semigroup becomes  the free monoid over Σ.
A probability distribution F together with all convolution  powers of F, with convolution as operation. This is called a convolution  semigroup.
A monoid is a semigroup  with an identity  element.
A group is a monoid in  which every element has an inverse  element.

Basic concepts
Identity and zero
Every semigroup, in fact every magma, has  at most one identity  element. A semigroup with identity is called a monoid. A semigroup  without identity may be embedded into a  monoid simply by adjoining an element  to  and defining  for all .  The notation S1 denotes a monoid obtained from S by  adjoining an identity if necessary (S1 = S for a monoid). Thus, every commutative semigroup can  be embedded in a group via the Grothendieck  group construction.
Similary, every magma has at most one absorbing  element, which in semigroup theory is called a zero.  Analogous to the above construction, for every semigroup S, one defines S0, a semigroup with 0 that embeds S.
Subsemigroups and ideals
The semigroup operation induces an operation on the  collection of its subsets: given subsets A and B of a semigroup, A*B, written  commonly as AB, is the set { ab |  a in A and b in B }. In terms of this  operations, a subset A is called

a subsemigroup if AA is a subset of A,
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).
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.
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.
Green's  relations, a set of five equivalence  relations that characterise the elements in terms of the principal  ideals they generate, are important tools for analysing the ideals of a  semigroup and related notions of structure.
Homomorphisms and congruences
A semigroup homomorphism is a function that preserves semigroup  structure. A function f: S → T between two semigroups is a homomorphism if the  equation
f(ab) = f(a)f(b).
holds for all elements a, b in S, i.e. the result is the same  when performing the semigroup operation after or before applying the map f. A semigroup homomorphism is not necessarily a monoid  homomorphism.
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). Isomorphic semigroups have the  same structure.
A semigroup congruence ˜ is an equivalence  relation that is compatible with the semigroup operation. That is, a subset   that is an equivalence relation and  and  implies  for every x,y,u,v in S. Like any equivalence relation, a semigroup  congruence ˜ induces congruence  classes

and the semigroup operation induces a binary operation  on the congruence classes:

Because ˜ is a  congruence, the set of all congruence classes of ˜ forms a semigroup with ,  called the quotient semigroup or factor semigroup, and denoted S / ˜. The mapping  is a semigroup homomorphism, called the quotient map, canonical surjection or projection; if S is a  monoid then quotient semigroup is a monoid with identity [1]˜. Conversely, the kernel of  any semigroup homomorphism is a semigroup congruence. These results are nothing  more than a particularization of the first  isomorphism theorem in universal algebra.
Every ideal I of a semigroup induces a  subsemigroup, the Rees  factor semigroup via the congruence x ρ y   ⇔   either x = y or both x and y are in I.
Structure of semigroups
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. If it is finite and nonempty, then it must contain at  least one idempotent. It  follows that every nonempty periodic semigroup has at least one idempotent.
A subsemigroup which is also a group is called a subgroup. 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.
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. For more on the structure of finite semigroups, see Krohn-Rhodes  theory.
Special classes of semigroups
Main article: Special classes of semigroups

A monoid is a semigroup  with identity.
A subsemigroup is a subset of a  semigroup that is closed under the semigroup operation.
A band is a  semigroup the operation of which is idempotent.
A cancellative semigroup is one having the cancellation  property:[1] a · b = a ·  c implies b = c and similarly for b · a = c · a.
Semilattices: A  semigroup whose operation is idempotent and commutative is  a semilattice.
0-simple semigroups.
Transformation semigroups: any finite semigroup S can be represented by transformations of a (state-) set Q of at most |S|+1 states. Each  element x of S then maps Q into itself x: Q → Q and sequence xy is defined by q(xy) = (qx)y for each q in Q. Sequencing  clearly is an associative operation, here equivalent to function  composition. This representation is basic for any automaton or finite  state machine (FSM).
The bicyclic  semigroup is in fact a monoid, which can be described as the free  semigroup on two generators p and q, under the relation p q = 1.
C0-semigroups.
Regular  semigroups. Every element x has at least one inverse  y satisfying xyx=x and yxy=y;  the elements x and y are sometimes  called "mutually inverse".
Inverse  semigroups are regular semigroups where every element has exactly one  inverse. Alternatively, a regular semigroup is inverse if and only if any two  idempotents commute.
Affine semigroup: a semigroup that is isomorphic to a  finitely-generated subsemigroup of Zd. These  semigroups have applications to commutative  algebra.

Group of fractions
The group of fractions of a semigroup  S is the group G = G(S) generated by the elements of S as generators and all equations xy=z which hold true in S as  relations.[2] This has a universal property for morphisms from S to a  group.[3] There is an obvious map from S to G(S) by sending each element of S to  the corresponding generator.
An important question is to characterize those semigroups for  which this map is an embedding. This need not always be the case: for example,  take S to be the semigroup of subsets of some set X with set-theoretic  intersection as the binary operation (this is an example of a semilattice).  Since A.A = A holds for all elements of S, this  must be true for all generators of G(S) as well: which is  therefore the trivial group.  It is clearly necessary for embeddability that S have the  cancellation  property. When S is commutative this condition is  also sufficient[4] and the Grothendieck  group of the semigroup provides a construction of the group of fractions.  The problem for non-commutative semigroups can be traced to the first  substantial paper on semigroups, (Suschkewitsch  1928).[5] Anatoly  Maltsev gave necessary and conditions for embeddability in 1937.[6]
Semigroup  methods in partial differential equations
Further information: C0-semigroup
Semigroup theory can be used to study some problems in the  field of partial differential equations. Roughly speaking, the semigroup  approach is to regard a time-dependent partial differential equation as an ordinary differential equation on a function space. For example,  consider the following initial/boundary value problem for the heat equation on the spatial interval (0, 1) ⊂ R and times t ≥ 0:

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

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

On an heuristic level, the solution to this problem "ought"  to be u(t) = exp(tA)u0.  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.
History
The study of semigroups trailed behind than that of other  algebraic structures with more complex axioms such as groups or rings. A number  of sources[7][8] 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.
Anton Suschkewitsch obtained 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.[8] From that point on, the foundations of semigroup theory were further laid by David  Rees, James  Alexander Green, Evgenii Sergeevich Lyapin, Alfred H.  Clifford and Gordon  Preston. The latter two published a two-volume monograph on semigroup theory  in 1961 and 1967 respectively. In 1970, a new periodical called Semigroup Forum (currently edited by Springer Verlag) became one of the few mathematical journals  devoted entirely to semigroup theory.
In recent years researchers in the field have became more  specialized with dedicated monographs appearing on important classes of  semigroups, like inverse  semigroups, as well as monographs focusing on applications in algebraic  automata theory, particularly for finite automata, and also in functional  analysis.
Generalizations

Group-like structures

Totality Associativity Identity Inverses

Group
Yes
Yes
Yes
Yes

Monoid
Yes
Yes
Yes
No

Semigroup
Yes
Yes
No
No

Loop
Yes
No
Yes
Yes

Quasigroup
Yes
No
No
No

Magma
Yes
No
No
No

Groupoid
No
Yes
Yes
Yes

Category
No
Yes
Yes
No

If the associativity axiom of a semigroup is dropped, the  result is a magma, which  is nothing more than a set M equipped with a binary  operation M × M → M.
Generalizing in a different direction, an n-ary semigroup (also n-semigroup, polyadic semigroup or multiary  semigroup) is a generalization of a semigroup to a set G with a n-ary operation instead of a binary operation.[9] The associative law is generalized as follows: ternary associativity is (abc)de = a(bcd)e = ab(cde), i.e. the string abcde with any three adjacent elements bracketed. N-ary  associativity is a string of length n + (n − 1) with any n adjacent  elements bracketed. A 2-ary semigroup is just a semigroup. Further axioms lead  to an n-ary group.
See  also

Absorbing  element
Biordered set
Empty  semigroup
Identity  element
Light's associativity test
Weak inverse

Notes

^ (Clifford  & Preston 1967, p. 3)
^ B. Farb, Problems on mapping class groups and related topics (Amer.  Math. Soc., 2006) page 357. ISBN 0821838385
^ M. Auslander  and D.A. Buchsbaum, Groups, rings, modules (Harper&Row, 1974) page 50. ISBN 006040378X
^ (Clifford  & Preston 1961, p. 34)
^ G. B. Preston (1990). "Personal reminiscences of the early  history of semigroups". http://www.gap-system.org/~history/Extras/Preston_semigroups.html. Retrieved  2009-05-12.
^ Maltsev, A. (1937), "On the immersion of an algebraic ring into a field", Math. Annalen 113: 686–691, doi:10.1007/BF01571659.
^ Earliest Known Uses of Some of the Words  of Mathematics
^ a b An account of Suschkewitsch's paper by  Christopher Hollings
^ Dudek, W.A. (2001), "On some old problems in n-ary  groups", Quasigroups and Related Systems 8: 15–36, http://www.quasigroups.eu/contents/contents8.php?m=trzeci

References
General references

Howie,  John M. (1995), Fundamentals of Semigroup Theory, Clarendon Press, ISBN 0-19-851194-9 .
Clifford, A.  H.; Preston, G.  B. (1961), The algebraic theory of semigroups, volume  1, American Mathematical Society .
Clifford, A.  H.; Preston, G.  B. (1967), The algebraic theory of semigroups, volume  2, American Mathematical Society .
Grillet, Pierre Antoine (1995), Semigroups:  an introduction to the structure theory, Marcel Dekker, Inc.

Specific references

Feller,  William (1971), An introduction to probability theory and  its applications. Vol. II., Second edition, New York: John Wiley &  Sons, MR0270403 .
Hille, Einar;  Phillips, Ralph S. (1974), Functional analysis and  semi-groups, Providence, R.I.: American Mathematical Society, MR0423094 .
Suschkewitsch, Anton (1928), "Über die endlichen Gruppen ohne  das Gesetz der eindeutigen Umkehrbarkeit", Mathematische Annalen 99 (1): 30–50,  doi:10.1007/BF01459084, MR1512437, ISSN 0025-5831 .
Kantorovitz, Shmuel (2010), Topics in  Operator Semigroups., Boston, MA: Birkhauser

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