Trees and spectra of Heyting algebras

A poset is Esakia representable when it is isomorphic to the prime spectrum of a Heyting algebra. Notably, every Esakia representable poset is also the spectrum of a commutative ring with unit. The problem of describing the Esakia representable posets was raised in 1985 and remains open to this day. We recall that a forest is a disjoint union of trees and that a root system is the order dual of a forest. It is shown that a root system is Esakia representable if and only if it satisfies a simple order theoretic condition, known as “having enough gaps”, and each of its nonempty chains has an infimum. This strengthens Lewis’s characterization of the root systems which are spectra of commutative rings with unit. While a similar characterization of arbitrary Esakia representable forests seems currently out of reach, we show that a well-ordered forest is Esakia representable if and only if it has enough gaps and each of its nonempty chains has a supremum.