I have the following task:
Prove that a semigroup $S$ is a rectangular band if and only if it is nowhere commutative.
So, if I get it right, what I have to prove has $2$ sides.
First, I need to prove that if I have a nowhere commutative semigroup, that must be a rectangular band.
Let us have a semigroup $S$ which is nowhere commutative. Let us have arbitrary $a,b \in S$. Then, if $ab=ba \rightarrow a=b$.
I need to prove that in that case, $aba =a$ $\forall a,b \in S$.
Second, I need to prove that if I have a rectangular band, then it must be a nowhere commutative semigroup.
Let us have a rectangular band $B$. In that case, $aba=a$. We need to prove that if $ab = ba \rightarrow a=b$.
I am new to algebra, and I have no idea how could one prove such things. I hope I asked the question right, any help appreciated. :)