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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?

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  • $\begingroup$ If you have a way to define associators, you can look at how much of your structure they (in some appropriate sense) generate. $\endgroup$ – Henning Makholm Jun 12 at 13:11
  • $\begingroup$ For relations, you have antitransitivity $\endgroup$ – Wojowu Jun 12 at 22:00
  • $\begingroup$ The Hasse diagram of a finite partial order might be considered minimally transitive. $\endgroup$ – Daniel Schepler Jun 12 at 22:02
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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.

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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.

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  • $\begingroup$ the empty relation would be "maximally intransitive" under your definition, even though it is transitive. Also, it is not uniquely defined. assume that $xRy$ hold for all $x,y\in \tilde X$ for some subset $\tilde X$. There are multiple relations satisfying the definition that are not isomorphic. Maybe we cannot get around that but it's not obvious to me. $\endgroup$ – user56834 Jun 13 at 8:11
  • $\begingroup$ @user56834. See edit. Maximal things usually are not unique. $\endgroup$ – William Elliot Jun 13 at 12:18
  • $\begingroup$ but this notion is not really sensible. like I said, there are relations that are "transitive" but "maximally intransitive" according to this definition. moreover, a "maximally intransitive" relation $R$ can be made more intransitive (in an intuitive sense), by adding elements to $R$. $\endgroup$ – user56834 Jun 13 at 18:27
  • $\begingroup$ @user56834. See revision. $\endgroup$ – William Elliot Jun 14 at 1:18
  • $\begingroup$ If $R$ is the empty relation, $T$ is transitive. If $R$ is the relation only containing $(a,b)$, $T$ is also transitive. If $R$ is a massive relation with lots of connections, removing only $(a,b)$ doesn’t make it “maximally intransitive” in any meaningful sense. $\endgroup$ – user56834 Jun 14 at 3:22

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