Algebra of Functions There seems to be an interesting algebra of functions. Does it already exist in literature?
Given functions $f_1 : X_1 \to Y$ and $f_2 : X_2 \to Y$, if $f_1(x) = f_2(x)$ for all $x \in X_1 \cap X_2$, then their sum is defined as $(f_1 + f_2) : X_1 \cup X_2 \to Y$, where
$$
(f_1 + f_2)(x) =
\begin{cases}
f_1(x) & x \in X_1\\
f_2(x) & x \in X_2.
\end{cases}
$$
Given functions $f : X \to Y$ and $g : Y \to Z$, their product is defined as $fg : X \to Z$, where
$$
(fg)(x) = g(f(x)).
$$
We immediately have the following:


*

*Addition is commutative and associative.

*The empty function with domain the empty set $\emptyset$ is the additive identity.

*Mutliplication is associative and distributes over addition.

*The identity function is the multiplicative identity.


These properties seem so nice that this algebra must have been investigated before - does anyone have some references for this?
 A: I think this setup is pretty interesting, but having addition and multiplication only partially defined does not make this into an algebra in the ring theoretic sense. (I guess you are using it in the universal algebra sense :) )
The general the feeling I get is that algebraic objects with partially defined operations are important in mathematical logic (which I know next to nothing about, really!) So, let me tell you about my luck searching.
A google search for ("partially defined" addition multiplication) immediately returned an ArXiV paper with the addition operation you mentioned on the first page. Another hit I got was this talk about "ringoids" which seems to discuss partially defined addition and multiplication.
Around this time I became convinced that "partial algebra" as the correct search term, but I had trouble finding an explanation that was both clear and trustworthy...until I found my answer in this paper by M.L. Reyes! He explains what I think you're looking for clearly, and he points us to Kochen and Specker in the references for more information (reference [11]).
I hope this helps. If I were you I'd start with Manny's paper.
