# examples of semigroups

Examples of semigroups are numerous. This entry presents some of the most common examples.

1. 1.

The set $\mathbb{Z}$ of integers with multiplication is a semigroup, along with many of its subsets (subsemigroups):

1. (a)

The set of non-negative integers

2. (b)

The set of positive integers

3. (c)

$n\mathbb{Z}$, the set of all integral multiples of an integer $n$

4. (d)

For any prime $p$, the set of $\{p^{i}\mid i\geq n\}$, where $n$ is a non-negative integer

5. (e)

The set of all composite integers

2. 2.

$\mathbb{Z}_{n}$, the set of all integers modulo an integer $n$, with integer multiplication modulo $n$. Here, we may find examples of nilpotent and idempotent elements, relative inverses, and eventually periodic elements:

1. (a)

If $n=p^{m}$, where $p$ is prime, then every non-zero element containing a factor of $p$ is nilpotent. For example, if $n=16$, then $6^{4}=0$.

2. (b)

If $n=2p$, where $p$ is an odd prime, then $p$ is a non-trivial idempotent element ($p^{2}=p$), and since $2^{p-1}\equiv 1\pmod{p}$ by Fermat’s little theorem, we see that $a=2^{p-2}$ is a relative inverse of $2$, as $2\cdot a\cdot 2=2$ and $a\cdot 2\cdot a=a$

3. (c)

If $n=2^{m}p$, where $p$ is an odd prime, and $m>1$, then $2$ is eventually periodic. For example, $n=96$, then $2^{2}=4$, $2^{3}=8$, $2^{4}=16$, $2^{5}=32$, $2^{6}=64$, $2^{7}=32$, $2^{8}=64$, etc…

3. 3.

The set $M_{n}(R)$ of $n\times n$ square matrices over a ring $R$, with matrix multiplication, is a semigroup. Unlike the previous two examples, $M_{n}(R)$ is not commutative.

4. 4.

The set $E(A)$ of functions on a set $A$, with functional composition, is a semigroup.

5. 5.

Every group is a semigroup, as well as every monoid.

6. 6.

If $R$ is a ring, then $R$ with the ring multiplication (ignoring addition) is a semigroup (with $0$).

7. 7.

Group with Zero. A semigroup $S$ is called a group with zero if it contains a zero element $0$, and $S-\{0\}$ is a subgroup of $S$. In $R$ in the previous example is a division ring, then $R$ with the ring multiplication is a group with zero. If $G$ is a group, by adjoining $G$ with an extra symbol $0$, and extending the domain of group multiplication $\cdot$ by defining $0\cdot a=a\cdot 0=0\cdot 0:=0$ for all $a\in G$, we get a group with zero $S=G\cup\{0\}$.

8. 8.

As mentioned earlier, every monoid is a semigroup. If $S$ is not a monoid, then it can be embedded in one: adjoin a symbol $1$ to $S$, and extend the semigroup multiplication $\cdot$ on $S$ by defining $1\cdot a=a\cdot 1=a$ and $1\cdot 1=1$, we get a monoid $M=S\cup\{1\}$ with multiplicative identity $1$. If $S$ is already a monoid with identity $1$, then adjoining $1^{\prime}$ to $S$ and repeating the remaining step above gives us a new monoid with identity $1^{\prime}$. However, $1$ is no longer an identity, as $1^{\prime}=1\cdot 1^{\prime}$.

Title examples of semigroups ExamplesOfSemigroups 2013-03-22 18:37:16 2013-03-22 18:37:16 CWoo (3771) CWoo (3771) 7 CWoo (3771) Example msc 20M99 group with 0 group with zero