Compositions of relations: is $\odot$ different from $\circ$ in it's usage?

Question

$\circ$ is the symbol used for compositions of functions. For example, the effect of $f \circ g$ is to "apply" $g$ first, then $f$.

Compositions of relations, on the other hand, are denoted with $\odot$ apparently. The effect of $R \odot S$ is to apply $R$ first, and then $S$ (the opposite of function compositions), and that's why the different notation ($\odot$ vs. $\circ$) is used.

But on this site for example I've seen only $\circ$ be used, even in the context of relations. I haven't been able to figure out whether $R \circ S$ is applying the relation $R$ first or the relation $S$ first.

Does the type of operator you use ($\odot$ or $\circ$) change which relation you apply first?

Answer 1

You should be prepared for any symbol to be used for binary operations. Within some areas of maths particular symbols tend to be used for particular compositions but this is not standardized across all mathematical disciplines.

In fact I am sure I can find books with $f \circ g$ meaning apply $f$ first.

Answer 2

S. Dolan suggests that notation is not standardized in mathematics. They are basically correct—there is no body in mathematics analogous to the Académie française (which seeks to standardize the French language).

That being said, there are some notations which are fairly standard across most of mathematics. In general, the more common or universally applicable a notation is, the more likely it is to mean (roughly) the same thing everywhere. For example, sets are a fairly fundamental part of all parts of mathematics, hence the notation for common set operations is basically universal: "$\cup$" almost certainly denotes a union, and "$\cap$" an intersection.

The use of "$\circ$" for functional composition is likewise nearly universal, as is the "backwards" right-to-left expansion. While I do not dispute S. Dolan's claim that there are books in which $(f \circ g)(x)$ expands to $g(f(x))$ (that is, apply $f$ first, then $g$), this is an unusual convention.^[1] Unless otherwise noted, it is safe to assume that $$ (f\circ g)(x) = f(g(x)). $$ While this is not a standard, it is a very common (nearly universal) convention.

On the other hand, composition of relations is not something that most mathematicians worry about: relations which are not functions just don't come up that often: I'm not sure that I have seen such relations outside of elementary courses on set theory (where they are introduced, then forgotten in favor of functions pretty quickly), and in a research seminar on applications of category theory to networks (a highly specialized branch of mathematics). Because general relations don't occur that often, the notation for composition is not very universal, and different authors likely adopt different notations. When reading a text in which relations are composed, it is important to pay attention to the conventions which the author(s) introduce. With respect to the question being asked here:

• "$\odot$" is not, so far as I know, a commonly understood notation for the composition of relations, and I have certainly seen it used to mean other things. For example, when I took my first course in modern algebra, the author of the text used $\oplus$ and $\odot$ to denote binary operations which would eventually be the addition and multiplication operations in a ring. These notations were used to distinguish, for example, integer multiplication from multiplication in $\mathbb{Z}/n\mathbb{Z}$.

• The choice of composition from left-to-right vs right-to-left will likely depend on the author of a text, as well. Of course, the "domain" and "codomain" of the relations in question should make it clear what is going on. That is, if $R \subseteq X \times Y$ and $S \subseteq Y\times Z$ are relations, and $\odot$ denotes composition of relations, then the only possible reading of $$ R \odot S $$ is that $R$ is applied first, then $S$ is applied second.

Aesop: The moral of the story is that notation can mean whatever the author intends it to mean. When in doubt, look for a definition and/or hope for an index of notation somewhere.

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[1] There is something aesthetically pleasing about interpreting things in the nonconventional order: given the diagram $$ X \xrightarrow{f} Y \xrightarrow{g} Z,$$ it seems reasonable to read from left-to-right, and use $f \circ g$ to denote the application of $f$ first, and then $g$. Indeed, given the prevalence of (commutative) diagrams in category theory and parts of algebra, I would imagine that the books S. Dolan has in mind are in those areas.

Fractal geometers also have some difficulty with the order of composition, but this occurs one step back: given a set of maps $\{ f_{k} \}_{k\in I}$ and a sequence $\alpha = (\alpha_1, \dotsc, \alpha_n)$ with each $\alpha_i \in I$, the notation $f_\alpha$ can mean either of $$ f_{\alpha} = f_{\alpha_1} \circ\dotsb\circ f_{\alpha_n} \qquad\text{or}\qquad f_{\alpha} = f_{\alpha_n} \circ\dotsb\circ f_{\alpha_1}, $$ depending on the author. This particular notational confusion is one of the most annoying bugaboos of my own work. :\