# 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:

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

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

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

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:

 $a=\left(\begin{matrix}1&0\\ 0&1\end{matrix}\right)\qquad b=\left(\begin{matrix}1&0\\ 0&0\end{matrix}\right)$
Title semigroup with two elements SemigroupWithTwoElements 2013-03-22 16:21:42 2013-03-22 16:21:42 rspuzio (6075) rspuzio (6075) 11 rspuzio (6075) Example msc 20M99