examples of semigroups

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

  1. 1.

    The set 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, the set of all integral multiplesMathworldPlanetmathPlanetmath of an integer n

    4. (d)

      For any prime p, the set of {piin}, where n is a non-negative integer

    5. (e)

      The set of all composite integers

  2. 2.

    n, the set of all integers modulo an integer n, with integer multiplication modulo n. Here, we may find examples of nilpotentPlanetmathPlanetmathPlanetmathPlanetmath and idempotent elements, relative inverses, and eventually periodic elements:

    1. (a)

      If n=pm, where p is prime, then every non-zero element containing a factor of p is nilpotent. For example, if n=16, then 64=0.

    2. (b)

      If n=2p, where p is an odd prime, then p is a non-trivial idempotent element (p2=p), and since 2p-11(modp) by Fermat’s little theorem, we see that a=2p-2 is a relative inverse of 2, as 2a2=2 and a2a=a

    3. (c)

      If n=2mp, where p is an odd prime, and m>1, then 2 is eventually periodic. For example, n=96, then 22=4, 23=8, 24=16, 25=32, 26=64, 27=32, 28=64, etc…

  3. 3.

    The set Mn(R) of n×n square matricesMathworldPlanetmath over a ring R, with matrix multiplicationMathworldPlanetmath, is a semigroup. Unlike the previous two examples, Mn(R) is not commutativePlanetmathPlanetmathPlanetmath.

  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 elementMathworldPlanetmath 0, and S-{0} is a subgroupMathworldPlanetmath 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 by defining 0a=a0=00:=0 for all aG, we get a group with zero S=G{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 on S by defining 1a=a1=a and 11=1, we get a monoid M=S{1} with multiplicative identity 1. If S is already a monoid with identityPlanetmathPlanetmath 1, then adjoining 1 to S and repeating the remaining step above gives us a new monoid with identity 1. However, 1 is no longer an identity, as 1=11.

Title examples of semigroups
Canonical name ExamplesOfSemigroups
Date of creation 2013-03-22 18:37:16
Last modified on 2013-03-22 18:37:16
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 7
Author CWoo (3771)
Entry type Example
Classification msc 20M99
Synonym group with 0
Defines group with zero