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?