I spent a lot of time trying to construct concrete example in the title.
What I got so far was - structure lack associativity or structure is monoid ( contains identity). I spent hours of searching the web. Usually authors state some very advanced claim or state structure which is of course monoid or group.
I have been offerd bicyclic semigroup - but this is monoid, so it doesnt meet the question.
Lack of my success and negative response on mathstack would suggest this doesn't exist. However I doubt there is not possible multiplication table.
Thank you all.
$B_2$ consists of 5 elements $e_{ij}, i,j=1,2$, and $0$. The product is $e_{i,j}e_{k,l}=0$ if $j\ne k$ and $=e_{i, l}$ otherwise. This is the 5-element Brandt semigroup. A 3-element inverse semigroup which is not a monoid is the semilattice $ \{a, b, 0\}$ with the product $aa=a, bb=b, ab=ba=0$ and $0$ is zero. Every 2- or 1- element inverse semigroup is a monoid.