Recently Ciabattoni, Galatos, and Terui have introduced a new notion of completion of residuated lattices which they call the hyper-MacNeille completion. This type of completion plays a central role in their work on establishing the admissibility of the cut-rule in certain types of hypersequent calculi for substructural logics.
In this talk we will focus on describing the hyper-MacNeille completion of Heyting algebras. We show that the hyper-MacNeille completion and the MacNeille completion coincide for so-called centrally supplemented Heyting algebras. That is, Heyting algebras with order-duals being Stone lattices. In fact, we show that any Heyting algebra A has a centrally supplemented extension S(A) such that the hyper-MacNeille completion of A is isomorphic to the MacNeille completion of S(A). These centrally supplemented extensions turn out to be the order-duals of the 1-Stone extensions originally introduced by Davey in 1972.
Time permitting, we will also discuss sheaf representations of centrally supplemented Heyting algebras and the problem of describing the MacNeille completions of lattices of global sections of a sheave of lattices over a Boolean space.
This is joint work with John Harding of New Mexico State University.
References:
[1] A. Ciabattoni, N. Galatos, and K. Terui, Algebraic proof theory: Hypersequents and hypercompletions. Ann. Pure Appl. Logic, 186(3) 693--737 (2017).
[2] G. D. Crown, J. Harding, and M. F. Janowitz, Boolean products of lattices. Order 13(2) 175--205 (1996).
[3] B. A. Davey m-Stone lattices, Canadian J. Math., 24(6) 1027--1032 (1972).
[4] J. Harding, Completions of orthomodular lattices II. Order 10(3) 283--294 (1993).