# What do you call a finite set of maps on $\mathbb{Z}$ that are closed and compatible with operations on $\mathbb{Z}$?

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.

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You have a set of maps, but what is the codomain? –  M Turgeon Mar 23 '12 at 17:45
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. –  Enjoys Math Mar 23 '12 at 17:51

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".
I used copier paper and a .5 mm mechanical pencil, but here's a feeble attempt with MathJax:$$\begin{array}{ccc}\mathbb Z^2 & \rightarrow^{\phi\times\psi} & X^2 \\ \downarrow \cdot && \downarrow * \\ \mathbb Z & \rightarrow^{\nu} & Y\end{array}$$ –  Henning Makholm Mar 23 '12 at 18:06
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). –  Henning Makholm Mar 23 '12 at 18:23