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:

$a\cdot a$ $=$ $a$

$a\cdot b$ $=$ $b$

$b\cdot a$ $=$ $b$

$b\cdot b$ $=$ $b$

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

$a\cdot (a\cdot a)=a\cdot a=$ $a$ $=a\cdot a=(a\cdot a)\cdot a$

$a\cdot (a\cdot b)=a\cdot b=$ $b$ $=a\cdot b=(a\cdot a)\cdot b$

$a\cdot (b\cdot b)=a\cdot b=$ $b$ $=b\cdot b=(a\cdot b)\cdot b$

$b\cdot (a\cdot a)=b\cdot a=$ $b$ $=a\cdot a=(a\cdot a)\cdot a$

$a\cdot (b\cdot b)=a\cdot b=$ $b$ $=b\cdot b=(a\cdot b)\cdot b$

$b\cdot (a\cdot b)=b\cdot b=$ $b$ $=b\cdot b=(b\cdot a)\cdot b$

$b\cdot (b\cdot a)=b\cdot b=$ $b$ $=b\cdot a=(b\cdot b)\cdot a$

$b\cdot (b\cdot b)=b\cdot b=$ $b$ $=b\cdot b=(b\cdot b)\cdot b$

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 $a\cdot b=b\cdot b$.

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 $\cdot$ to be the logical connective ”and”, we obtain this semigroup in logic. We may also represent it by matrices like so:

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