Egli-Milner order

Let (A, \sqsubseteq) be a partial order.

The order on A induces two preorders on the powerset \mathcal{P}(A), the so-called lower and upper preorders:

\begin{aligned}
X \sqsubseteq_\text{l} Y &\equiv \forall x \in X.\ \exists y \in Y.\ x \sqsubseteq y \\
X \sqsubseteq_\text{u} Y &\equiv \forall y \in Y.\ \exists x \in X.\ x \sqsubseteq y
\end{aligned}

Used together, these induce the Egli-Milner preorder:

  X \sqsubseteq_\text{EM} Y \equiv
  	X \sqsubseteq_\text{l} Y
  	\land 
    X \sqsubseteq_\text{u} Y