Semigroup

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 . 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 .

FORMAL DEFINITION

A semigroup formally consists of a pair

(S,\cdot_S)

where ''S'' is a set and a binary function

\cdot_S: S 	imes S 
ightarrow 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

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 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

S^1

) 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 .

EXAMPLES OF SEMIGROUPS

Any monoid, and therefore any Group .

The positive Integer s with addition.

Any Subset of a semigroup closed under the semigroup operation. Such a subset is known as a subsemigroup.

A semigroup whose operation is Idempotent is a Band .

A semigroup whose operation is Idempotent and Commutative is a Semilattice .

Any Ideal of a Ring , given multiplication. Thus any ring including the Integer s, rational, real, complex or Quaternion ic numbers, functions with values in a ring (including sequences), Polynomial s and Matrices .

--- Matrix Unit s form a 0-simple semigroup.

--- Square Nonnegative Matrices with matrix multiplication.

The set of all finite Strings over some fixed alphabet Σ, with string concatenation as operation. If the empty string is included, then this is actually a monoid, called the " Free Monoid over Σ"; if it is excluded, then we have a semigroup, called the " Free Semigroup over Σ".

A . This representation is basic for any Automaton or Finite State Machine (FSM).

The Bicyclic Semigroup .

C₀-semigroups .

Regular Semigroup s.

Inverse Semigroup s.

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 this case, ''T'' and ''S'' are also isomorphic, and for the purposes of semigroup theory, the two semigroups are identical.

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Semigroup