In mathematics, the bicyclic semigroup is an algebraic object important for the structure theory of semigroups.

The bicyclic semigroup has the property that the image of any morphism φ from B to another semigroup S is either cyclic, or it is an isomorphic copy of B.

As the name implies, free monoids and semigroups are those objects which satisfy the usual universal property defining free objects, in the respective categories of monoids and semigroups.

Each free semigroup (or monoid) S has exactly one set of free generators, the cardinality of which is called the rank of S. As the name implies, free monoids and semigroups are those objects which satisfy the usual universal property defining free objects, in the respective categories of monoids and semigroups.

$43.00 (C) In this volume the authors represent the leading areas of research in semigroup theory and its applications, both to other areas of mathematics and to areas outside mathematics.

The editors include papers that survey Clifford's work on Clifford semigroups and trace the influence of Clifford's work on current semigroup theory.

Also notable is a paper on applications from other areas of mathematics to semigroup theory, and a paper on an application of semigroup theory to theoretical computer science and mathematical logic.

In particular, a regular semigroup with one idempotent is a group: as such, many interesting subclasses of regular semigroups arise from putting conditions on the idempotents.

This is version 19 of regular semigroup, born on 2004-06-04, modified 2006-10-04.

The free semigroup on A is the subsemigroup of A* containing all elements except the empty string.

Each free semigroup (or monoid) S has exactly one set of free generators, the cardinality of which is called the rank of S.

As the name implies, free monoids and semigroups are those objects which satisfy the usual universal property defining free objects, in the respective categories of monoids and 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).

If a monogenic semigroup is infinite then it is isomorphic to the semigroup of positive integers with the operation of addition.

MATHS: Semigroups(Site not responding. Last check: 2007-11-07)

A semigroup that has a commutative operation (so that a combined with b is the same as b combined with a) is said to be Abelian.

A common example of a semigroup is the free semigroup generated by a set of atomic elements A by concatenating them.

If f is a morphism from the semigroup S1 to S2 then it is a map and so define a partition of S1.Set into a collection of sets which themselves form a semigroup with an operation defined by the inverses image of the operation in the second semigroup S2.op.

[No title](Site not responding. Last check: 2007-11-07)

When applied to S, its value should be the set of elements of S. The carrier of semigroup(S,*) is S. multiplication This symbol represents a unary function, whose argument should be a semigroup S. It returns the multiplication map on S. We allow for the map to be n-ary.

Its first argument should be a semigroup S and the second and third arguments should be elements of S.

Its first argument should be a semigroup G. The second should be an arithmetic expression A, whose operators are times and power, and whose leaves are members of the carrier of G. The second argument of power should be positive.

Other important computations include determining the size of the semigroup, determining whether the semigroup is regular, finding certain subgroups in the semigroup and finding the congruence lattice of the semigroup.

Computations with semigroups in which the multiplication and comparison of elements can be described using automata is currently implied as with "coset enumeration" but is not done explicitly.

A semigroup library would similarly provide specific constructions of known classes of semigroups for which the Green's relations and other computations can be completed effectively.

Finitely presented semigroups are obtained by factoring a free semigroup by a set of relations (a generating set for the congruence), ie, a set of pairs of words in the free semigroup.

Such calculations comparing elements of an finitely presented semigroup may run into problems: there are finitely presented semigroups for which no algorithm exists (it is known that no such algorithm can exist) that will tell for two arbitrary words in the generators whether the corresponding elements in the finitely presented semigroup are equal.

If a finitely presented semigroup has a confluent rewriting system then it has a solvable word problem, that is, there is an algorithm to decide when two words in the free underlying semigroup represent the same element of the finitely presented semigroup.

Quotient semigroup presentations in misere impartial combinatorial games Thane Plambeck http://www.plambeck.org/ 15 Aug 2004 INTRODUCTION In these notes, we describe a natural quotient semigroup structure on the positions of an impartial game with fixed rules.

This is a four element semigroup (in fact it is more than that---a group, not just a semigroup).

The elements of the misere quotient semigroup are 1 (ie, the empty position), a, b, b^2, ab, and ab^2.

Semigroup and Automata Theory Home Page(Site not responding. Last check: 2007-11-07)

A good example of a semigroup is provided by the set of all binarystrings; any two such strings can be composed by concatenation to form a third binarystring, an operation which is clearly associative.

In mathematical terms, the semigroup of all binarystrings is the free semigroup on two generators, subsets of the free monoid are languages, and the computer is an example of a finite state machine.

For regular semigroups S, it is often the case that if all the local submonoids of S belong to some class C, then S can be covered by a Rees matrix semigroup over an element of C, where Rees matrix semigroups are the analogues of matrix rings.

If the set and corresponding table do not form a semigroup, your program should report that the pair do not form a semigroup and state why.

If the set and operation pair do form a semigroup, your program should check to see if the semigroup is also a commutativesemigroup.

In the first three results you should substitute actual elements of the set that yield a counter-example to the definitions for a semigroup and a commutative operation.

acm.uva.es /p/v3/398.html (904 words)

[No title](Site not responding. Last check: 2007-11-07)

semigroup3 http://www.openmath.org/cd/semigroup3.ocd 2006-06-01 2004-06-01 3 1 experimental Semigroup constructions Initiated by Arjeh M. Cohen 2003-10-02 cyclic_semigroup This symbol denotes the cyclic semigroup with a cycle of length l and a tail of length k.

When applied to X, it refers to the semigroup of all functions from X to X if X is a set and to {1,...,X} if X is an integer, whose binary operation is composition of maps and whose identity element is the identity map on the set X, respectively {1,...,X}.

direct_power This is a binary function whose first argument should be a semigroup M and whose second argument should be a natural number n.

Semigroups (algebras of time) and their expansions (algebras of histories) are applied to problems of historical grounding and story-telling for situated agents.

Introduction Expansions, which arose as useful techniques the decomposition theory of semigroups, are functorial ways of enlarging each semigroup S in the category of semigroups to an expanded one b S while providing a natural transformation back onto the original semigroup S. Semigroups are models of time, and expansions can...

Nehaniv, C., and Dautenhahn, K. Semigroup expansions for autobiographic agents.

citeseer.ist.psu.edu /50158.html (517 words)

Semigroup Theory(Site not responding. Last check: 2007-11-07)

Conference on Semigroups, Acts and Categories with Applications to Graphs, to celebrate the 65th birthdays of Mati Kilp and Ulrich Knauer, June 27-30, 2007, Tartu, ESTONIA.

A Conference on Representations of Algebras, Groups and Semigroups December 30, 2007-January 4, 2008 Ramat Gan and Netanya, Israel.

In case you don't have the Semigroup Forum macros, here they are :