1
$\begingroup$

Let $S$ be the set of maps and $\phi,\psi \in S$. Let $x,y \in \mathbb{Z}$. Suppose that $\phi(x) * \psi(y) = \nu(xy)$ for some $\nu \in S$. Then what would you call such a system of maps?

If that was easy, then what would you call such as system if all of the maps are only partially defined on $\mathbb{Z}$?

Note: $xy$ is the integer product, and $*$ is some binary operator on $\{\phi(x) : x \in \mathbb{Z}, \phi\in S\}$

Grazie.

  • 0
    Not known yet. If it helps, let the codomain be the same for each. But it might turn out to be different for each map, so $*$ might be considered a collection of binary operators. It's up in the air right now.2012-03-23

1 Answers 1

1

The triple $(\phi:\mathbb Z\to X,\psi:\mathbb Z\to X,\nu:\mathbb Z\to Y)$ describes a morphism from $(\cdot):\mathbb Z^2\to \mathbb Z$ to $(*):X^2\to Y$ in the arrow category over $\mathbf{Set}$. It probably doesn't have any short name, if you don't want to call it just a "commuting square".

If the maps are partial, it's the same thing, but over the category of sets and partial functions.

  • 0
    It's just a `\begin{array}...\end{array}` with arrows in some of the cells. You can right-click on any rendered formula on the site to see the source markup. (Though for some browsers you have to find a way to dismiss the browser right-click menu that appears on top of the MathJax menu).2012-03-23