abstract-spectral-characterization

Let $L$ be any complete lattice and let $a\mapsto a^\perp$ be a function $L\to L$ satisfying, for all relevant elements, $a\wedge a^\perp=\bot$, $a\vee a^\perp=\top$, $(a\wedge b)^\perp=a^\perp\vee b^\perp$, $(a\vee b)^\perp=a^\perp\wedge b^\perp$, and $(a^\perp)^\perp=a$, together with the additional law $a\wedge(b\vee c)=(a\wedge b)\vee(a\wedge c)$ whenever $b\le a$ and $c\le a^\perp$. A proper dual ideal $J\subseteq L$ means a nonempty set excluding $\bot$, upward closed, and closed under binary meets. Give the set $\mathcal D(L)$ of proper dual ideals the topology generated by the sets $\{J:a\in J\}$, for $a\in L$. For every function $f:\mathcal D(L)\to\mathbb R$, assume that $f$ is upper semicontinuous in this topology and that, for every nonempty family $\mathcal F\subseteq\mathcal D(L)$ and every $J\in\mathcal D(L)$ whose underlying set is $\bigcap_{K\in\mathcal F}K$, one has $f(J)=\sup\{f(K):K\in\mathcal F\}$. This supremum is the total real supremum operation, which returns $0$ for a set unbounded above; no boundedness hypothesis is imposed on this family of values. Then there exists exactly one function $E:\mathbb R\to L$ satisfying all of the following: $E$ is monotone; $E(t)=\inf_{s>t}E(s)$ for every real $t$; $\inf_{t\in\mathbb R}E(t)=\bot$ and $\sup_{t\in\mathbb R}E(t)=\top$; there exist real numbers $a,b$ such that $E(t)=\bot$ for every $t<a$ and $E(t)=\top$ for every $t>b$; and, for every proper dual ideal $J$,

The uniqueness is among functions satisfying these spectral conditions and this equality; no condition $a\le b$ is explicitly required of the two witnesses.