# Associative binary operations defined on a finite set

We can easily test the commutativity of a binary operation on a set of $\ n\in\mathbb{N}\$ elements by observing the symmetry of the $\ n\times n\$ multiplication table.

I was looking for a similar test for the associativity of a binary operation. Since it should work on "triples", I don't expect to find anything in a square table.

I tried with a $\ n\times n\times n\$ table, only in the case of binary operations on a set of two elements (I can only use a pencil), and it turned out that there are cubic tables for associative operations equal to cubic tables for non-associative operations.

So, do you happen to know a "model" to represent a binary operation from which it is clearly visible the associativity of the operation? Thanks!

• Light's associativity test is too expensive. There are some subsets from the multiplication table from which you are able to deduce the whole of it: you can find nice tricks from the thesis of Andreas Distler, just google this name and you wil find it Jan 5, 2014 at 7:10
• Thanks. My aim, in the first place, was to work out the number of associative binary operations that may be defined on a finite set. A lecturer of mine asked to do it for a set of 2 elements more than one year ago, and few months ago another teacher suggested to write a programm which could work out the number of associative binary operations for sets of three, four and perhaps five elments. The (slow) idea was to find every single operation and test its associavity. I don't think I'm able to use Light's test to shorten the time of the computing, but I hope to find something in this thesis. Jan 6, 2014 at 16:02
• the questions your teachers asked you to do is exactly what Andreas Distler does in his thesis. You should avoid such teachers who give you to solve the well-known to be solved problems Jan 7, 2014 at 7:09

See $\S 1.2$ "Light's associativity test" in: