What is the algebraic structure of sets with a “splitting” operation?

Question

I noticed that any convex shape can be split along a straight line and produce two shapes that are also convex. The general pattern seems like the dual of a semigroup—what is it called? A cosemigroup?

Answer 1

Before defining a "cosemigroup", you should define precisely the underlying semigroup and it is not really clear in your example.

That being said, if you start from a monoid $M$, you may consider the function $\tau$ from $M$ into $\mathcal{P}(M \times M)$, the set of subsets of $M \times M$, defined by: $$ \tau(x) = \bigl\{(x_1, x_2) \in M \times M \mid x_1x_2 = x \bigr\} $$ Thus $\tau(x)$ is the set of all possible decompositions of $x$ as a product of two elements of $x$. This map $\tau$ has been extensively studied when $M$ is a free monoid, where it leads to several algebraic developments. Let me know if you want to know more in this direction.