For some binary operation defined on a set, it is of interest whether the operation is associative or not (among, of course many other things).
Is there a measure (and accompanying theory) that describes the extent to which a binary operation is associative? For example, if this measure is $M$, then we could have $M(*, S) = 0$ if the $*$ defined on $S$ has 0 triplets $(a, b, c)$ for which $a * (b * c) = (a * b) * c$, and $M(*, S) = 1$ if the operation is associative (so for all triplets $a * (b * c) = (a * b) * c$). For operations where some triplets associate, we could assign a value between 0 and 1.
I wonder, for instance, how other properties will affect this measure. For example, a operation with identity has at least $7n + 1$ triplets which "associate" (with $n$ being the number of elements in the set $S$). A commutative operation will have even more.
I tried to do a Google search, but lacking any useful keywords I could not find anything.
Background: This question arose from another question where I asked for some help to prove that a particular operation is associative, where it turns out the operation is not associative at all. It surprised me for two reasons: I had some theoretical reasons (now proven to be wrong) to believe it, but also, in trying to get a proof I worked through several examples that where triplets were associating.
