Infinite associativity condition

Question

How associativity condition may be formulated for a function taking an arbitrary ordinal number of arguments?

For a binary operation $\ast$ it is $(a\ast b)\ast c = a\ast (b\ast c)$, but I want an infinite formula, whose special case is the condition of associativity for a binary operation.

Let's look at the next case: four args. What would the rules be? The possibilities are abcd = a(b(cd)) = a((bc)d) = (ab)(cd) = (a(bc)*d) = ... Seems to me that the only reasonable way which avoids enumerating all possible parenthesizations (not a word, but ...) would be a recursive one. — marty cohen, Mar 18 '12 at 19:21
There isn't really a notion of associativity when starting with a ternary operator, so an operator taking infinitely many variables probably doesn't have a real notion of associativity. — Thomas Andrews, Mar 18 '12 at 19:36
@Thomas: Writing $[abc]$ for the result of applying the operation to the ordered triple $\langle a,b,c\rangle$, the obvious notion of associativity is that $[ab[cde]]=[a[bcd]e]=[[abc]de]$ for all $a,b,c,d,e$. A much less obvious generalization of associativity is $[[abc]de]=[a[bde][cde]]$, obtained by thinking of each element $a$ as a two-place function (where the ordinary $(ab)c=a(bc)$ corresponds to thinking of each element as a one-place function). — Brian M. Scott, Mar 18 '12 at 22:57
See a related question at MathOverflow: http://mathoverflow.net/questions/91761/formula-for-a-product-of-products-over-ordinal-numbers — porton, Mar 20 '12 at 22:37
@BrianM.Scott I think you've provided useful information, but this doesn't contradict what Thomas Andrews says. Your generalization of the associativity, I agree, generalizes our intuitive concept of associativity to a non-binary operation. Associativity (once given a notational system) gets defined as an equality which involves a binary operation, which is not just an intuition. Hence you may claim to have a concept of ternary associativity, but that is not the concept of associativity, or redundantly, binary associativity. — Doug Spoonwood, Apr 11 '12 at 23:33
@Doug: I disagree. I would say that there is no generally accepted definition of associativity for ternary operations, not that associativity is necessarily a property of binary operations. — Brian M. Scott, Apr 12 '12 at 00:34
@BrianM.Scott If associativity is not a property of a binary operation, what in the world do you mean by associativity? If we have an equation like (a(bc))=((ab)c), how is * not a binary operation when it behaves exactly like a binary operation? — Doug Spoonwood, Apr 12 '12 at 01:47
@Doug: Of course in that example $$ is a binary operation: that’s an illustration of the accepted notion of associativity for binary operations. It also seems to me to be a non sequitur to my previous comment. Does it bother you that the term ideal* has been extended to settings very different from its original home? My view is that to date there simply hasn’t been any good reason to extend associativity to $n$-ary operations for $n\ne 2$ (and indeed there may never be). You seem to want to rule out any such extension a priori; not being a seer, I decline to do so, that’s all. — Brian M. Scott, Apr 12 '12 at 01:58
@BrianM.Scott I have no problems with extensions of terms. In fact, I like them. Using the term "associativity" for extensions of binary associativity isn't necessarily problematic, but you've then used the term "associativity" in a different way for the extended concept than you did for binary associativity. The meaning of the term would not be the same in both cases. — Doug Spoonwood, Apr 12 '12 at 02:10
@Doug: Nor is it for ideals. And in fact I suggested two ways to extend it to ternary operations, one very obviously a straightforward generalization of the binary property. I see no profit in pursuing the disagreement further, however; I’ve said what I have to say on the subject. — Brian M. Scott, Apr 12 '12 at 02:18
@BrianM.Scott Yes, you did. Sorry if it seemed like I implied otherwise. — Doug Spoonwood, Apr 12 '12 at 03:12

score 5 · Answer 1 · answered Mar 18 '12 at 20:04

For operations of arbitrary arity you should look at Monads. They conceptualize the notion of associativity to a rather abstract and general context.

For some ideas concerning infinite products see the note "Some remarks on infinite products" by Jim Coykendal, online

Infinite associativity condition

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