Is there a notion of "maximally intransitive" relation, or "maximally nonassociative" operator?

Question

Transitivity on relations $R\subseteq X\times X$ and associativity on binary operators $+:X\times X \to X$ are defined as:

$$\forall x,y,z, \quad xRy\land yRz\to xRz$$

$$\forall x,y,z, \quad (x+y)+z=x+(y+z)$$

I am interested whether there is some notion of a "most non-transitive" relation or "most non-associative" operator. Some seem to be more intransitive or more nonassociative than others.

If we define the formulas $T_R(t):xRy\land yRz\to xRz$ for $t=(x,y,z)$ and $A_+(t):(x+y)+z=x+(y+z)$, then it would be tempting to just say that the minimally transitive relation is the one for which this doesnt satisfy for any $t\in X^3$, but there clearly isnt such a relation. Similar for associativity.

Is there some way that we can formalize the notion of "minimal" transitivity/associativity?

Answer 1

Let S be a set with at least 3 points.
T = SxS is a transitive relation.
For distinct a,b, let R = T - {(a,b)}.
R is not transitive because there is a point c
with aRc and cRb but without aRb.
Thus R is maximally intransitive.

Answer 2

You could try to generalize the notion of center $Z(X)$ of a group ( or ring, or etc.) $X$ - namely, the set of elements which commute with everything. Of course, since associativity takes three elements into account, this is a bit stranger; I suspect that the right thing to look at, given a binary function $*$, is the set $$A(X)=\{b: \forall a,c[(a*b)*c=a*(b*c)]\}.$$ Note that $A(X)$ is itself closed under $*$ (this is easy to check) and is itself completely associative, which does suggest that this isn't too terrible a notion.

"Minimal associativity" then could mean that $A(X)$ is as small as possible, in whatever context we're considering. E.g. in an arbitrary magma this would mean $A(X)=\emptyset$; if we commit ourselves to having an identity element $e$, then $A(X)$ must contain it (since $(a*e)*c=a*c=a*(e*c)$) but could in principle consist of just the identity.

Unfortunately, while smallness of center is often important, I'm not aware of any situations where $A(X)$ being "small" is actually interesting. But then I'm not familiar with non-associative algebraic structures in general.