Strong set order

Real numbers

Consider the real numbers ${\displaystyle \mathbb {R} }$ with the usual order ${\displaystyle \geq }$. Let ${\displaystyle A=[{\underline {a}},{\overline {a}}]}$ and ${\displaystyle B=[{\underline {b}},{\overline {b}}]}$. Then

${\displaystyle A\leq _{s}B\iff {\underline {a}}\leq {\underline {b}}\ {\text{ and }}\ {\overline {a}}\leq {\overline {b}}}$.

More generally, for any ${\displaystyle A,B\subseteq \mathbb {R} }$, we have

${\displaystyle A\leq _{s}B\iff \inf A\leq \inf B\ {\text{ and }}\ \sup A\leq \sup B}$.

Euclidean space

Consider the Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ with the usual pointwise order: ${\displaystyle \mathbf {x} \leq \mathbf {y} \iff x_{i}\leq y_{i}\ \ \forall i\leq n}$. For ${\displaystyle n=2}$, let

${\displaystyle A=[0,2]\times [0,3],}$

${\displaystyle B=[1,3]\times [1,4].}$

Then :${\displaystyle A\leq _{s}B}$. But now consider

${\displaystyle B'=[-1,3]\times [1,4]}$;

Then ${\displaystyle A\nleq _{s}B'}$, since ${\displaystyle (0,0)\in A}$, ${\displaystyle (-1,1)\in B'}$ but

${\displaystyle (0,0)\wedge (-1,1)=(-1,0)\notin B'}$.

Power set

Consider the power set of the natural numbers ${\displaystyle {\mathcal {P}}(\mathbb {N} )}$ under the set-inclusion order: ${\displaystyle A\leq B\iff A\subseteq B}$. Let

${\displaystyle A=\{\{1\}\}}$,

${\displaystyle B=\{\{1\},\{2\},\{1,2,3\}\}}$.

It it not true that ${\displaystyle A\leq _{s}B}$. Indeed, we have ${\displaystyle \{1\}\in A}$, ${\displaystyle \{2\}\in B}$, but

${\displaystyle \{1\}\wedge \{2\}=\{1\}\cap \{2\}=\varnothing \notin A,}$

whence ${\displaystyle A\nleq _{s}B}$.

Topkis' Theorem

The strong set order is widely used in monotone comparative statics, in particular via Topkis' Theorem. Given a lattice ${\displaystyle X}$, a poset ${\displaystyle \Theta }$, a constraint correspondence ${\displaystyle D:\Theta \rightrightarrows X}$ and a function ${\displaystyle f:X\times \Theta \to \mathbb {R} }$, the theorem gives sufficient conditions for the correspondence

${\displaystyle x^{*}(\theta )=\arg \max _{x\in D(\theta )}f(x,\theta )}$

the be increasing in the strong set order, that is, for the statement

${\displaystyle \theta \geq \theta '\implies x^{*}(\theta )\geq _{s}x^{*}(\theta ')}$

to hold.

Monotone selections

Increasigness in the strong set order can be used to obtain monotone selections from correspondences. Indeed, if ${\displaystyle X}$ is a poset and ${\displaystyle Y}$ is a lattice, the correspondence ${\displaystyle \Gamma :X\rightrightarrows Y}$ is nonempty-valued and increasing in the strong set order, then a monotone selection ${\displaystyle g}$ from ${\displaystyle \Gamma }$ exists if any of the following hold:

• ${\displaystyle \Gamma (x)\subseteq Y}$ has a minimal element for every ${\displaystyle x\in X}$ (in particular, if ${\displaystyle \Gamma }$ is complete-lattice-valued). One can thus take :${\displaystyle g:X\to Y}$ by putting ${\displaystyle g(x)=\min \Gamma (x)}$. The same works if a maximal element is available.

• ${\displaystyle Y}$ is a sublattice of a finite product of chains. This covers, for example, ${\displaystyle Y\subseteq \mathbb {R} ^{n}}$.

• ${\displaystyle X}$ is countable.

Zhou's Fixed-Point Theorem for correspondences

The strong set order can also be used for a generalization of Tarski's fixed-point theorem to correspondences known as Zhou's fixed-point theorem:

Theorem: let ${\displaystyle X}$ be a nonempty complete lattice and ${\displaystyle \Gamma :X\rightrightarrows X}$ a nonempty-valued correspondence. If ${\displaystyle \Gamma }$ is increasing in the strong set order and ${\displaystyle \Gamma (x)}$ is a subcomplete sublattice for all ${\displaystyle x\in X}$, then ${\displaystyle \Gamma }$ has a fixed point. Moreover, the set ${\displaystyle {\text{Fix}}(\Gamma )}$ of such fixed points is a complete lattice.