semigroup with two elements
Perhaps the simplest non-trivial example of a semigroup which is not a group is a particular semigroup with two elements. The underlying set of this
semigroup is {a,b} and the operation
is defined as follows:
| a⋅a | = | a | ||
| a⋅b | = | b | ||
| b⋅a | = | b | ||
| b⋅b | = | b |
It is rather easy to check that this operation is associative, as it should be:
| a⋅(a⋅a)=a⋅a= | a | =a⋅a=(a⋅a)⋅a | ||
| a⋅(a⋅b)=a⋅b= | b | =a⋅b=(a⋅a)⋅b | ||
| a⋅(b⋅b)=a⋅b= | b | =b⋅b=(a⋅b)⋅b | ||
| b⋅(a⋅a)=b⋅a= | b | =a⋅a=(a⋅a)⋅a | ||
| a⋅(b⋅b)=a⋅b= | b | =b⋅b=(a⋅b)⋅b | ||
| b⋅(a⋅b)=b⋅b= | b | =b⋅b=(b⋅a)⋅b | ||
| b⋅(b⋅a)=b⋅b= | b | =b⋅a=(b⋅b)⋅a | ||
| b⋅(b⋅b)=b⋅b= | b | =b⋅b=(b⋅b)⋅b |
It is worth noting that this semigroup is commutative and has an identity
element
, which is a. It is not a group because the element b does
not have an inverse
. In fact, it is not even a cancellative semigroup
because we cannot cancel the b in the equation a⋅b=b⋅b.
This semigroup also arises in various contexts. For instance, if we choose a to be the truth value ”true” and b to be the truth value ”false” and the operation ⋅ to be the logical connective ”and”, we obtain this semigroup in logic. We may also represent it by matrices like so:
| a=(1001) |
| Title | semigroup with two elements |
|---|---|
| Canonical name | SemigroupWithTwoElements |
| Date of creation | 2013-03-22 16:21:42 |
| Last modified on | 2013-03-22 16:21:42 |
| Owner | rspuzio (6075) |
| Last modified by | rspuzio (6075) |
| Numerical id | 11 |
| Author | rspuzio (6075) |
| Entry type | Example |
| Classification | msc 20M99 |