4
$\begingroup$

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?

  • 0
    Is this isomorphic to anything we know?2012-06-28
  • 2
    On which underlying set of functions exactly is this algebra structure supposed to be defined? It looks like the multiplication is only partially defined.2012-06-28
  • 0
    @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.2012-06-28
  • 0
    @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.2012-06-28
  • 0
    multiplication is not distributive over addition on the right.2012-06-28
  • 0
    @mercio: Oh yes, sorry for overlooking that, thanks for pointing that out :)2012-06-28
  • 0
    @mercio Do you have an example of distributivity failing? I can't seem to see what would go wrong.2012-06-28
  • 0
    @rschwieb Using the previous definitions of $f_1$ and $f_2$, and given any $e : W \to (X_1 \cup X_2)$, the function $e(f_1 + f_2)$ exists but $ef_1$ and $ef_2$ both exist only when $f_1 = f_2$, which is not very interesting.2012-06-28
  • 0
    @HerngYi Well the philosophy here is that *being undefined is not a problem*. The point is that all of the axioms hold when all of the expressions are defined. (This appears in the papers I referred to.) To really contradict distributivity, we'd have to produce an example where everything is defined, but distributivity fails.2012-06-28
  • 0
    oh I'm sorry I didn't read your definition of addition. It should work when everything is defined. However sometimes, f*h+g*h (resp. f*g+f*h) is defined while (f+g)*h (resp. f*(g+h)) is not. And if you were to extend your definition of + to any pair of functions by saying undefined where they disagree, you also get subtle problems.2012-06-28
  • 0
    @Herng: Your "addition" (which should not be called addition) is known as a (special case of) pushout. Of course it is known.2012-06-28
  • 0
    @Martin, the essence (as "induced" by the interpretation in the context of functions) of the algebra above is in the distributivity of multiplication over addition, so I'm looking for a structure in literature which includes this algebra as a whole.2012-06-28

1 Answers 1

5

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.

  • 0
    Let me add that there he is focused on such objects with commutative multiplication, but he alludes that the noncommutative version exists as well.2012-06-28
  • 0
    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.2012-06-28
  • 0
    @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!2012-06-28
  • 0
    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...2013-05-29