examples of semigroups
Examples of semigroups are numerous. This entry presents some of the most common examples.
The set of non-negative integers
The set of positive integers
For any prime , the set of , where is a non-negative integer
The set of all composite integers
, the set of all integers modulo an integer , with integer multiplication modulo . Here, we may find examples of nilpotent and idempotent elements, relative inverses, and eventually periodic elements:
If , where is prime, then every non-zero element containing a factor of is nilpotent. For example, if , then .
If , where is an odd prime, then is a non-trivial idempotent element (), and since by Fermat’s little theorem, we see that is a relative inverse of , as and
If , where is an odd prime, and , then is eventually periodic. For example, , then , , , , , , , etc…
The set of functions on a set , with functional composition, is a semigroup.
Every group is a semigroup, as well as every monoid.
Group with Zero. A semigroup is called a group with zero if it contains a zero element , and is a subgroup of . In in the previous example is a division ring, then with the ring multiplication is a group with zero. If is a group, by adjoining with an extra symbol , and extending the domain of group multiplication by defining for all , we get a group with zero .
As mentioned earlier, every monoid is a semigroup. If is not a monoid, then it can be embedded in one: adjoin a symbol to , and extend the semigroup multiplication on by defining and , we get a monoid with multiplicative identity . If is already a monoid with identity , then adjoining to and repeating the remaining step above gives us a new monoid with identity . However, is no longer an identity, as .
|Title||examples of semigroups|
|Date of creation||2013-03-22 18:37:16|
|Last modified on||2013-03-22 18:37:16|
|Last modified by||CWoo (3771)|
|Synonym||group with 0|
|Defines||group with zero|