Let (A,⊑)(A, \sqsubseteq)(A,⊑) be a partial order.
The order on AAA induces two preorders on the powerset P(A)\mathcal{P}(A)P(A), the so-called lower and upper preorders:
X⊑lY≡∀x∈X. ∃y∈Y. x⊑yX⊑uY≡∀y∈Y. ∃x∈X. x⊑y\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} X⊑lYX⊑uY≡∀x∈X. ∃y∈Y. x⊑y≡∀y∈Y. ∃x∈X. x⊑y
Used together, these induce the Egli-Milner preorder:
X⊑EMY≡X⊑lY∧X⊑uY X \sqsubseteq_\text{EM} Y \equiv X \sqsubseteq_\text{l} Y \land X \sqsubseteq_\text{u} Y X⊑EMY≡X⊑lY∧X⊑uY