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 operationMathworldPlanetmath is defined as follows:

aa = a
ab = b
ba = b
bb = b

It is rather easy to check that this operation is associative, as it should be:

a(aa)=aa= a =aa=(aa)a
a(ab)=ab= b =ab=(aa)b
a(bb)=ab= b =bb=(ab)b
b(aa)=ba= b =aa=(aa)a
a(bb)=ab= b =bb=(ab)b
b(ab)=bb= b =bb=(ba)b
b(ba)=bb= b =ba=(bb)a
b(bb)=bb= b =bb=(bb)b

It is worth noting that this semigroup is commutativePlanetmathPlanetmathPlanetmath and has an identity elementMathworldPlanetmath, which is a. It is not a group because the element b does not have an inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath. In fact, it is not even a cancellative semigroup because we cannot cancel the b in the equation ab=bb.

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)  b=(1000)
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