What Does the Associative Property Mean Intuitively Across All Notational Schemes?

Question

You can find descriptions of associativity as intuitively meaning that the order of operations performed does not matter, e. g. such as that of Wikipedia. However, if you write what associativity means in terms of a formula in either prefix notation, that is that for some binary operation X, XXxyz=XxXyz, or suffix notation xyzXX=xyXzX, the intuitive description of associativity loses sense. So, what does associativity mean intuitively in a prefix or suffix scheme? I suppose one might say that in prefix and suffix notation, associativity means that one can push the second instance of the operation in as far as we can, or as far out as we can and still have an equivalent expression. But, this does not seem to fit with an intuitive description of associativity in an infix notation.

So, what does associativity mean across all notational schemes? Can we meaningfully talk about associativity across all notational schemes, or do intuitive descriptions only work as local to a particular notational scheme?

Addendum: I'm not quite sure if this belongs here or on Philosophy Stack Exchange. I would like it imported there, if it seems more fitting there.

Answer 1

I disagree with the premise implicit in the question that the meaning of associativity depends on the notational scheme used to denote operations. Associativity means that the order of operations doesn't matter (or that left and right multiplication commute, as Bill pointed out). What this description loses when you use prefix or postfix notation is not sense but visibility. The meaning of natural numbers as a universal counting device doesn't lose sense in Japanese just because it uses several dozen counting particles for different classes of things.

If, on the other hand, the question were how to describe syntactically how associativity manifests in various notational schemes, I'd say the answer is that in infix notation parentheses become unnecessary, in prefix notation you can write the operations as early as you want to (and no later than their operands) and in postfix notation you can write them as late as you want to (and no earlier than their operands). Whereas in infix notation there's no "canonical" form for a multiple operation (i.e. no reason to prefer one of $x\circ(y\circ z)$ and $(x\circ y)\circ z$ over the other), and the clearest way to write one is by dropping the parentheses, in prefix/postfix notation the canonical form would seem to be to have all the operators at the front/end.

Answer 2

The nature of associativity can be best realized by realizing that a binary operator:

$$b:X\times X \rightarrow X$$

Has a natural adjoint (where $X^X$ is the set of functions from $X$ to $X$:)

$$b^*:X \rightarrow X^X$$

defined by: $b^*(x)(y)=b(x,y)$

A binary operation is then associative if and only if it corresponds to function composition on $X^X$. That is, there is a natural composition binary operator:

$$\circ: X^X \times X^X \rightarrow X^X$$

Then $b$ is associative if and only if for all $x,y\in X$:

$$b^*(b(x,y)) = b^*(x) \circ b^*(y)$$

So, in that sense, associativity is always represented as composition of functions.

Comment added much later:

As Joriki noted in comments, there is another adjoint, $^*b$, which is defined as $^*b(x)(y)=b(y,x)$.

In some sense, then, prefix notation, $Kxy$, can be thought of as representing $Kx=b^*(x)$ applied to $y$. And $xyK$ can be seen as the operation $^*b(y)$ to x. In that sense, the prefix notation represents the first adjoint, $b^*$, the infix represents the binary operator, $b$, and the suffix notation represents the second adjoint, $^*b$.

Answer 3

An alternative point on view of associativity is that it isn’t about binary operations at all.

An associative operation on a set is, intuitively, a way to aggregate any finite tuple of elements from the set into an element of the set — formally, a family of operations $\mu_n : X^n \to X$, for each $n>0$ (or possibly $n \geq 0$ — more on this later) — such that any way of combining these operations, without permutation, deletion, or duplication, is the same as the specified operation of that length.

For instance: given a list $(x,y,z,w)$, we could take $\mu_4(x,y,z,w)$ directly; or we could look at the composite operation $\mu_2(x,\mu_3(y,z,w))$. (Notice: no permutation, duplication, or deletion of the elements $(x,y,z,w)$.) Generalised associativity says that these are equal.

The classical associativity law, $\mu_2(\mu_2(x,y),z) = \mu_2(x,\mu_2(y,z))$ clearly follows from this (since both are equal to $\mu_3(x,y,z)$; and it so happens that given $\mu_2$ satisfying this, one can always reconstruct the rest of the operations. So one can get away with just working with binary operations and this single law. However, intuitively, I find it useful to think of associative operations as really being the whole family of $n$-ary operators, for all $n > 0$. (Taking $n \geq 0$ corresponds to associative and unital binary operations.)

In practice, for instance, if I want to check whether some operation is associative, the first thing I’ll do is to try writing down a formula for a generalised $n$-ary version; if this comes naturally, then the operation will almost always be associative.

For example, the convolution product for functions from a finite group into a ring is defined by $$ (f \ast g)(x) := \sum_{yz = x} f(y) g(z) $$ Calculating associativity by hand takes maybe three lines and forty-five seconds — but you can see immediately, just by looking at it, that it generalises to a ternary version, $$ (f \ast g \ast h)(x) := \sum_{yzw = x} f(y) g(z) h(w) $$ (and to $n$-ary versions for other $n$), and this gives a strong heuristic clue that it will be associative.

This viewpoint is formalised in the theory of operads, and is essential for generalising to subtler notions such as “up-to-homotopy-associative”.