Definition:Semigroup

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Definition

Let $\struct {S, \circ}$ be a magma.


Then $\struct {S, \circ}$ is a semigroup if and only if $\circ$ is associative on $S$.


That is:

A semigroup is an algebraic structure which is closed and whose operation is associative.


Semigroup Axioms

The properties that define a semigroup can be gathered together as follows:


A semigroup is an algebraic structure $\struct {S, \circ}$ which satisfies the following properties:

\((\text S 0)\)   $:$   Closure      \(\ds \forall a, b \in S:\) \(\ds a \circ b \in S \)      
\((\text S 1)\)   $:$   Associativity      \(\ds \forall a, b, c \in S:\) \(\ds a \circ \paren {b \circ c} = \paren {a \circ b} \circ c \)      


Multiplicative Semigroup

Let $\struct {S, \circ}$ be a semigroup whose operation is multiplication.


Then $\struct {S, \circ}$ is a multiplicative semigroup.


Additive Semigroup

Let $\struct {S, \circ}$ be a semigroup whose operation is addition.


Then $\struct {S, \circ}$ is an additive semigroup.


Also defined as

Some sources specify that a semigroup must be non-empty, thus denying the possibility of $S = \O$ for such a structure.


Also known as

Some older texts have this as semi-group.

A semigroup is also known as an associative algebraic structure.

Some sources call this a monoid, but this term usually has a different meaning.

Make sure you understand which is being used.


Examples

Operation Defined as $x + y + x y$ on Positive Integers

Let $\circ: \Z_{\ge 0} \times \Z_{\ge 0}$ be the operation defined on the integers $\Z_{\ge 0}$ as:

$\forall x, y \in \Z_{\ge 0}: x \circ y := x + y + x y$

Then $\struct {\Z_{\ge 0}, \circ}$ is a semigroup.


Operation Defined as $x + y - x y$ on Integers

Let $\circ: \Z \times \Z$ be the operation defined on the integers $\Z$ as:

$\forall x, y \in \Z: x \circ y := x + y - x y$

Then $\struct {\Z, \circ}$ is a semigroup.


Order $2$ Semigroups

The Cayley tables for the complete set of semigroups of order $2$ are listed below.

The underlying set in all cases is $\set {a, b}$.

$\begin{array}{r|rr}
 & a & b \\

\hline a & a & a \\ b & a & a \\ \end{array} \qquad \begin{array}{r|rr} 

 & a & b \\

\hline a & a & a \\ b & a & b \\ \end{array} \qquad \begin{array}{r|rr} 

 & a & b \\

\hline a & a & a \\ b & b & b \\ \end{array}$

$\begin{array}{r|rr}
 & a & b \\

\hline a & a & b \\ b & a & b \\ \end{array} \qquad \begin{array}{r|rr} 

 & a & b \\

\hline a & a & b \\ b & b & a \\ \end{array} \qquad \begin{array}{r|rr} 

 & a & b \\

\hline a & a & b \\ b & b & b \\ \end{array}$

$\begin{array}{r|rr}
 & a & b \\

\hline a & b & a \\ b & a & b \\ \end{array}$

$\begin{array}{r|rr}
 & a & b \\

\hline a & b & b \\ b & b & b \\ \end{array}$


Also see

  • Results about semigroups can be found here.


Sources

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