This chart shows that inverse semigroups, have associativity and divisibility, but not invertibility.
At first this made me think that they might not have invertibility despite the word "inverse" being right there in the name. But after reading its Wikipedia article I found that they do in fact have invertibility.
So my question is, why does this chart not show invertibility for inverse semigroups when it shows invertibility for other structures like loops and groups.
Is the chart wrong/incomplete for the sake of looking nice? What would the correct chart look like? What other inaccuracies are in this chart?
The issue is that invertibility in a semigroup means something different than it does in groups and monoids. Groups and monoids have an identity so we define inverses based on that. However semigroups don't have an identity element so invertibility in that context means that for any $x$ there exists a unique $y$ such that $xyx=x$. This property doesn't allow us to do left or right cancellations in the same way we can in groups so it's kept distinct. The diagram suggests that groups are in the intersection of invertible semigroups and monoids and this is true so it's in the right spot.