Is there a standard name/notation for this partial order over subsets of $\mathbb{R}^{n+1}$?

Question

Let consider $\mathbb{R}^{n+1}$ and let us denote with $x_1, \dots, x_n, y$ the usual coordinates on $\mathbb{R}^{n+1}$. We will also use the notation $x = (x_1, \dots, x_n)$. Moreover let us denote by $L_x$ the "vertical" line passing through $(x_1, \dots, x_n, 0)$, i.e. $$ L_x := \{(x_1, \dots, x_n, t) : t \in \mathbb{R}\}. $$

Let us consider the following partial order $\lhd $ over the power set $\mathcal{P}(\mathbb{R}^{n+1})$.

For $A, B \subset \mathbb{R}^{n+1}$: $$ A \lhd B \qquad \text{iff} \qquad \forall x \in \mathbb{R}^n: (L_x \cap A \ne \emptyset \, \,\wedge \,\, L_x \cap B\ne \emptyset) \rightarrow (L_x\cap A \le L_x \cap B) $$ where "$L_x\cap A \le L_x \cap B$" is to be meant in the sense of subsets of $\mathbb{R}$.

So in other words, $A \lhd B$ iff $B$ lies "above" $A$.

Is there a standard name and/or notation for $\lhd$?

Answer 1

That relation is not a partial order. Consider the following subsets of $\mathbb R^2$: $$A=\{(1,0),(3,1)\}$$ $$B=\{(1,1),(2,0)\}$$ $$C=\{(2,1),(3,0)\}$$ Then $A\lhd B\lhd C\lhd A$ and $A\not\lhd C,$ so the relation $\lhd$ is not transitive.