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

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?

• There is a notion of co-(insert any algebraic structure here), but this is not what happens for convex shapes, since there is no canonical choice of the line along which you cut the shape. – Martin Brandenburg May 7 '15 at 9:49
• @MartinBrandenburg: You’re right. So if I introduced some arbitrary canonical choice—always split a shape in half along the x-axis, always split a list at its midpoint, &c.—then this would be a cosemigroup? – Jon Purdy May 7 '15 at 17:06

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.