44 research outputs found
Sheaf representations of MV-algebras and lattice-ordered abelian groups via duality
We study representations of MV-algebras -- equivalently, unital
lattice-ordered abelian groups -- through the lens of Stone-Priestley duality,
using canonical extensions as an essential tool. Specifically, the theory of
canonical extensions implies that the (Stone-Priestley) dual spaces of
MV-algebras carry the structure of topological partial commutative ordered
semigroups. We use this structure to obtain two different decompositions of
such spaces, one indexed over the prime MV-spectrum, the other over the maximal
MV-spectrum. These decompositions yield sheaf representations of MV-algebras,
using a new and purely duality-theoretic result that relates certain sheaf
representations of distributive lattices to decompositions of their dual
spaces. Importantly, the proofs of the MV-algebraic representation theorems
that we obtain in this way are distinguished from the existing work on this
topic by the following features: (1) we use only basic algebraic facts about
MV-algebras; (2) we show that the two aforementioned sheaf representations are
special cases of a common result, with potential for generalizations; and (3)
we show that these results are strongly related to the structure of the
Stone-Priestley duals of MV-algebras. In addition, using our analysis of these
decompositions, we prove that MV-algebras with isomorphic underlying lattices
have homeomorphic maximal MV-spectra. This result is an MV-algebraic
generalization of a classical theorem by Kaplansky stating that two compact
Hausdorff spaces are homeomorphic if, and only if, the lattices of continuous
[0, 1]-valued functions on the spaces are isomorphic.Comment: 36 pages, 1 tabl
Bi-intermediate logics of trees and co-trees
A bi-Heyting algebra validates the G\"odel-Dummett axiom iff the poset of its prime filters is a disjoint union of co-trees (i.e.,
order duals of trees). Bi-Heyting algebras of this kind are called bi-G\"odel
algebras and form a variety that algebraizes the extension
- of bi-intuitionistic logic axiomatized by the
G\"odel-Dummett axiom. In this paper we initiate the study of the lattice
- of extensions of
-.
We develop the methods of Jankov-style formulas for bi-G\"odel algebras and
use them to prove that there are exactly continuum many extensions of
-. We also show that all these extensions can be
uniformly axiomatized by canonical formulas. Our main result is a
characterization of the locally tabular extensions of
-. We introduce a sequence of co-trees, called the
finite combs, and show that a logic in - is locally
tabular iff it contains at least one of the Jankov formulas associated with the
finite combs. It follows that there exists the greatest non-locally tabular
extension of - and consequently, a unique pre-locally
tabular extension of -. These results contrast with
the case of the intermediate logic axiomatized by the G\"odel-Dummett axiom,
which is known to have only countably many extensions, all of which are locally
tabular
A representation of odd Sugihara chains via weakening relations
We present a relational representation of odd Sugihara chains. The elements
of the algebra are represented as weakening relations over a particular poset
which consists of two densely embedded copies of the rationals. Our
construction mimics that of Maddux (2010) where a relational representation of
the even Sugihara chains is given. An order automorphism between the two copies
of the rationals is the key to ensuring that the identity element of the monoid
is fixed by the involution.Comment: 14 pages, 1 figur
A categorical approach to the maximum theorem
Berge's maximum theorem gives conditions ensuring the continuity of an
optimised function as a parameter changes. In this paper we state and prove the
maximum theorem in terms of the theory of monoidal topology and the theory of
double categories.
This approach allows us to generalise (the main assertion of) the maximum
theorem, which is classically stated for topological spaces, to
pseudotopological spaces and pretopological spaces, as well as to closure
spaces, approach spaces and probabilistic approach spaces, amongst others. As a
part of this we prove a generalisation of the extreme value theorem.Comment: 45 pages. Minor changes in v2: this is the final preprint for
publication in JPA
Percolation on self-dual polygon configurations
Recently, Scullard and Ziff noticed that a broad class of planar percolation
models are self-dual under a simple condition that, in a parametrized version
of such a model, reduces to a single equation. They state that the solution of
the resulting equation gives the critical point. However, just as in the
classical case of bond percolation on the square lattice, self-duality is
simply the starting point: the mathematical difficulty is precisely showing
that self-duality implies criticality. Here we do so for a generalization of
the models considered by Scullard and Ziff. In these models, the states of the
bonds need not be independent; furthermore, increasing events need not be
positively correlated, so new techniques are needed in the analysis. The main
new ingredients are a generalization of Harris's Lemma to products of partially
ordered sets, and a new proof of a type of Russo-Seymour-Welsh Lemma with
minimal symmetry assumptions.Comment: Expanded; 73 pages, 24 figure