A binary relation $R$ on a set $S$ is
- transitive if $\forall a, b, c \in S: aRb \wedge bRc \Rightarrow aR$c,
- semi-transitive if $\forall a, b, c, d \in S: aRb \wedge bRc \Rightarrow (aRd \vee dRc)$. (see [1,2])
Any transitive relation is also semi-transitive.
Let $\mathbf{u}, \mathbf{v} \in \mathbb{R}^n$. The point $\mathbf{u}$ Pareto-dominates $\mathbf{v}$ denoted by $\mathbf{u} \succeq \mathbf{v}$ if $\forall i \in [n]: u_i \leq v_i$ and $\exists j \in [n]: u_j < v_j$. Evidently, Pareto-domination is transitive.
Now let $\mathbf{t} = (0,0)$, $\mathbf{u} = (1,1)$, $\mathbf{v} = (2,2)$ and $\mathbf{w} = (-1,10)$. We get that $\mathbf{t} \succeq \mathbf{u}$, $\mathbf{u} \succeq \mathbf{v}$, however neither $\mathbf{t} \succeq \mathbf{w}$ nor $\mathbf{t} \succeq \mathbf{v}$, so Pareto-domination is not semi-transitive. (In fact, $\mathbf{w}$ is non-dominated w.r.t. $\mathbf{t}, \mathbf{u}, \mathbf{v}$.)
Question: How is it possible that transitivity implies semi-transitivity, while a transitive relation such as Pareto domination on $\mathbb{R}^n$ is not semi-transitive?
What am I missing here? Is the definition of semi-transitivity incomplete?