Maximal d-spectra and locally compact Hausdorff spaces

Abstract

It is an interesting open problem whether every compact Hausdorff space can be realized as the maximal $d$-spectrum of an arithmetic frame. We approach this problem by generalizing the $d$-nucleus to a stably continuous frame. We use Priestley duality to characterize the resulting $\underline d$-nucleus, which allows us to prove that every locally compact Hausdorff space can be realized as the maximal $\underline d$-spectrum of a continuous regular frame. As a corollary, we obtain that every locally Stone space can be realized as the maximal $d$-spectrum of an algebraic regular frame.