Semigroup without left cancellation law and right cancellation law

Question

Can please someone give example of a Semigroup in which neither left cancellation law hold nor right cancellation law

Answer 1

Any nontrivial ring (e.g. integers or reals) with multiplication works (because for any $x,y$ we have $0\cdot x=x\cdot 0=y\cdot 0=0\cdot y$).

Answer 2

A finite commutative monoid of order - e.g. - $4$ that makes the job is

$$\begin{array}{c|cccc} \ast & 1 & 2 & 3 & 4\\\hline 1 & 1 & 2 & 3 & 4\\ 2 & 2 & 2 & 3 & 4\\ 3 & 3 & 3 & 3 & 4\\ 4 & 4 & 4 & 4 & 4\\ \end{array}$$ Here, for example, $$\color{red}{2}\ast 1=\color{red}{2}\ast 2\quad\text{but}\quad 1\ne 2$$ and $$1\ast \color{red}{4}=3\ast \color{red}{4}\quad\text{but}\quad 1\ne 3.$$