of $T$-transitivity has been put forward [@DeBaets2002; @Moser2006].

A symmetric relation $Q: {\mathcal{V}}^2 \rightarrow [0,1]$ is called $T$-transitive with $T$ a t-norm if for any $({v}_1,{v}_2,{v}_3) \in {\mathcal{V}}^3$ $$\begin{aligned}
T(Q({v}_1,{v}_2),Q({v}_2,{v}_3)) \leq
Q({v}_1,{v}_3)\,.\end{aligned}$$

Three important t-norms are the minimum t-norm $T_{\bf M}(a,b) = \min(a,b)$, the product t-norm $T_{\bf P}(a,b) = ab$ and the Łukasiewicz t-norm $T_{\bf
L}(a,b) = \max(a+b-1,0)$.

In addition, several authors have shown that various forms of transitivity give rise to utility representable or numerically representable relations, also called fuzzy weak orders – see e.g. [@Luce1965; @Billot1995; @Koppen1995; @Fono2007; @Bodenhofer2007]. We will use the term ranking representability to establish a link with machine learning. We give a slightly specific definition that unifies reciprocal and symmetric relations.

A reciprocal or symmetric relation ${Q}: {\mathcal{V}}^2 \rightarrow [0,1]$ is called ranking representable if there exists a ranking function ${f}: {\mathcal{V}}\rightarrow {\mathbb{R}}$ such that for all $({v},{v}')
\in {\mathcal{V}}^2$ it respectively holds that

${Q}({v},{v}') = \nabla ({f}({v}) - {f}({v}')) \,$ (reciprocal case) ;

${Q}({v},{v}') = \nabla ({f}({v}) + {f}({v}')) \,$ (symmetric case) .

The main idea is that ranking representable relations can be constructed from a utility function $f$. Ranking representable reciprocal relations correspond to directed acyclic graphs, and a unique ranking of 