${\mathcal{V}}^2$ to $\mathbb{R}$. If $K^\phi$ on ${\mathcal{V}}$ is universal, then for every function $t\in S({\mathcal{V}}^2)$ and every $\epsilon > 0$, there exists a function $h$ in the RKHS (\[eq:primalmodel\]) induced by the kernel (\[eq:recedgekernel\]), such that $$\max_{({v},{v}')\in {\mathcal{V}}^2}\left\{\left\arrowvert t({v},{v}')-{h}({v},{v}')\right\arrowvert\right\}\leq\epsilon.$$

The proof is analogous to that of Theorem \[antisymmetrictheorem\] (see appendix).

As a side note, we remark that a symmetric and reciprocal version of the Cartesian kernel can be introduced as well.

Relationships with fuzzy set theory
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The previous section revealed that specific Kronecker product edge kernels can be constructed for modeling reciprocal and symmetric relations, without requiring any further background about these relations. In this section we demonstrate that the Kronecker product edge kernels $K_{\otimes}^{\Phi}$, $K_{\otimes R}^{\Phi}$ and $K_{\otimes S}^{\Phi}$ are particularly useful for modeling intransitive relations. Intransitive relations occur in a lot of real-world scenarios, like game playing [@DeSchuymer2003; @Fisher2008], competition between bacteria [@Kerr2002; @czaran2002chemical; @nowak2002; @Kirkup2004; @karolyi2005rps; @reichenbach2007] and fungi [@Boddy2000], mating choice of lizards [@Sinervo1996] and food choice of birds [@Waite2001], to name just a few. In an informal way, Figure \[fig:examples\] shows with the help of examples what transitivity means for symmetric and reciprocal relations that are crisp and graded.

Despite the occurrence of intransitive 