I am reading Section 2.4.1 in Boyd's Convex Optimization, where the proper cone and the generalized inequality are defined. Specifically, given proper cone $\mathcal{K}\subset \mathcal{R}^n$ and $x$, $y\in\mathcal{R}^{n}$, $x\preceq_{\mathcal{K}}y\Leftrightarrow y-x\in\mathcal{K}$.
Transitivity is not included in the list of properties of strict generalized inequality (Page 44).
May I ask why strict generalized inequality is not transitive? Could you please introduce a counterexample to me?