arXiv subjects

M. A. Moshier

Publications and source records attributed to M. A. Moshier.

3 recordsLinked to original sources

arXiv2026

Maximal d-spectra and locally compact Hausdorff spaces

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.

math.GN↗

arXiv2025

Dedekind-MacNeille and related completions: subfitness, regularity, and Booleanness

Completions play an important rôle for studying structure by supplying elements that in some sense ``ought to be." Among these, the Dedekind-MacNeille completion is of particular importance. In 1968 Janowitz provided necessary and sufficient conditions for it to be subfit or Boolean. Another natural separation axiom connected to these is regularity. We explore similar characterizations of when closely related completions are subfit, regular, or Boolean. We are mainly interested in the Bruns-Lakser, ideal, and canonical completions, which (unlike the Dedekind-MacNeille completion) satisfy stronger forms of distributivity. The first two are widely used in pointfree topology, while the latter is of crucial importance in the semantics of modal logic.

math.GN↗

arXiv2025

Degrees of join-distributivity via Bruns-Lakser towers

We utilize the Bruns-Lakser completion to introduce Bruns-Lakser towers of a meet-semilattice. This machinery enables us to develop various hierarchies inside the class of bounded distributive lattices, which measure $κ$-degrees of distributivity of bounded distributive lattices and their Dedekind-MacNeille completions. We also use Priestley duality to obtain a dual characterization of the resulting hierarchies. Among other things, this yields a natural generalization of Esakia's representation of Heyting lattices to proHeyting lattices.

math.LO↗