# 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:

1. Addition is commutative and associative.
2. The empty function with domain the empty set $\emptyset$ is the additive identity.
3. Mutliplication is associative and distributes over addition.
4. 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?

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Is this isomorphic to anything we know? –  akkkk Jun 28 '12 at 8:50
On which underlying set of functions exactly is this algebra structure supposed to be defined? It looks like the multiplication is only partially defined. –  Rasmus Jun 28 '12 at 9:00
@Auke: I'm an amateur so I know very little literature. I'm sure it's isomorphic to something we know, though, I'm just hoping someone could tell me what it is. –  Herng Yi Jun 28 '12 at 9:14
@Rasmus: I guess one could restrict the domains and codomains of the functions to give a more "complete" structure, but it seems interesting enough even when multiplication is partially defined. –  Herng Yi Jun 28 '12 at 9:14
multiplication is not distributive over addition on the right. –  mercio Jun 28 '12 at 12:28

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.

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Let me add that there he is focused on such objects with commutative multiplication, but he alludes that the noncommutative version exists as well. –  rschwieb Jun 28 '12 at 12:51
He discusses much about the generalized class of such algebraic structures, but I am interested in this particular example involving functions. I am actually working on a project that defines a "partial algebra" that is an extension of this algebra of functions. However, I don't know any literature so I was hoping to find out how much has been covered previously. –  Herng Yi Jun 28 '12 at 13:06
@HerngYi It's possible there are papers more specific than what I gave, but in my experience, the stuff I found above is a good level of generality to look at. Good luck on your project in any case! –  rschwieb Jun 28 '12 at 13:27
sorry for necromancing but I'm still working on this... Reyes's paper says "The notion of a partial algebra was deﬁned in [11, §2]." Well that citation is a paper on Quantum Mechanics and mentions no partial algebra... –  Herng Yi May 29 '13 at 14:32