Topological duality and lattice expansions, II: Lattice expansions with quasioperators

Topological Duality and Lattice Expansions Part II: Lattice Expansions with Quasioperators

Lattices have many applications in mathematics and logic, in which they occur together with additional operations. For example, in applications of Hilbert spaces, one is often concerned with the lattice of closed subspaces of a fixed space. This lattice is not distributive, but there is an operation taking a given subspace to its orthogonal subspace. More generally, ortholattices are lattices w...

Topological Duality and Lattice Expansions Part I: A Topological Construction of Canonical Extensions

The two main objectives of this paper are (a) to prove topological duality theorems for semilattices and bounded lattices, and (b) to show that the topological duality from (a) provides a construction of canonical extensions of bounded lattices. The paper is the first of two parts. The main objective of the sequel [MJ2] is to establish a characterization of lattice expansions, i.e., lattices wi...

MacNeille completions of lattice expansions

There are two natural ways to extend an arbitrary map between (the carriers of) two lattices, to a map between their MacNeille completions. In this paper we investigate which properties of lattice maps are preserved under these constructions, and for which kind of maps the two extensions coincide. Our perspective involves a number of topologies on lattice completions, including the Scott topolo...

Two theorems on lattice expansions

Zone determinant expansions for nuclear lattice simulations

We consider quantum simulations of nuclear matter on the lattice. In particular, we address the problem of calculating the contribution of nucleon/nucleon-hole loops at nonzero nucleon density. With the help of auxiliary boson fields, all nucleon interactions can be written in terms of one-body interactions in a fluctuating background. In the grand canonical ensemble, the contribution of nucleo...