relations in many domains, one has to admit that most applications are still characterized by relations that fulfill relatively strong transitivity requirements. For example, in decision making, preference modeling and social choice theory, one can argue that reciprocal relations like choice probabilities and preference judgments should satisfy certain transitivity properties, if they represent rational human decisions made after well-reasoned comparisons on objects [@Luce1965; @Fishburn1991; @Tversky1998]. For symmetric relations as well, transitivity plays an important role [@Gower1986; @Jakel2008], when modeling similarity relations, metrics, kernels, etc.

It is for this reason that transitivity properties have been studied extensively in fuzzy set theory and related fields. For reciprocal relations, one traditionally uses the notion of stochastic transitivity [@Luce1965].

Let $g$ be an increasing $[1/2,1]^2 \rightarrow [0,1]$ mapping. A reciprocal relation $Q: {\mathcal{V}}^2 \rightarrow [0,1]$ is called $g$-stochastic transitive if for any $({v}_1,{v}_2,{v}_3)
\in {\mathcal{V}}^3$ $$\begin{aligned}
\big( Q({v}_1,{v}_2) \geq 1/2 \wedge Q({v}_2,{v}_3) \geq 1/2 \big)
\Rightarrow Q({v}_1,{v}_3) \geq
g(Q({v}_1,{v}_2),Q({v}_2,{v}_3))\,.\end{aligned}$$

Important special cases are weak stochastic transitivity when $g(a,b) =
1/2$, moderate stochastic transitivity when $g(a,b) =
\min(a,b)$ and strong stochastic transitivity when $g(a,b) =
\max(a,b)$. Alternative (and more general) frameworks are FG-transitivity [@Switalski2003] and cycle transitivity [@DeBaets2005; @DeBaets2006]. For graded symmetric relations, the notion 