12 research outputs found

### Dedekind sigma-complete l-groups and Riesz spaces as varieties

We prove that the category of Dedekind $\sigma$-complete Riesz spaces is an
infinitary variety, and we provide an explicit equational axiomatization. In
fact, we show that finitely many axioms suffice over the usual equational
axiomatization of Riesz spaces. Our main result is that $\mathbb{R}$, regarded
as a Dedekind $\sigma$-complete Riesz space, generates this category as a
quasi-variety, and therefore as a variety. Analogous results are established
for the categories of (i) Dedekind $\sigma$-complete Riesz spaces with a weak
order unit, (ii) Dedekind $\sigma$-complete lattice-ordered groups, and (iii)
Dedekind $\sigma$-complete lattice-ordered groups with a weak order unit.Comment: 15 page

### The dual of compact ordered spaces is a variety

In a recent paper (2018), D. Hofmann, R. Neves and P. Nora proved that the
dual of the category of compact partially ordered spaces and monotone
continuous maps is a quasi-variety - not finitary, but bounded by $\aleph_1$.
An open question was: is it also a variety? We show that the answer is
affirmative. We describe the variety by means of a set of finitary operations,
together with an operation of countably infinite arity, and equational axioms.
The dual equivalence is induced by the dualizing object [0,1]

### Operations that preserve integrability, and truncated Riesz spaces

For any real number $p\in [1,+\infty)$, we characterise the operations
$\mathbb{R}^I \to \mathbb{R}$ that preserve $p$-integrability, i.e., the
operations under which, for every measure $\mu$, the set $\mathcal{L}^p(\mu)$
is closed. We investigate the infinitary variety of algebras whose operations
are exactly such functions. It turns out that this variety coincides with the
category of Dedekind $\sigma$-complete truncated Riesz spaces, where truncation
is meant in the sense of R. N. Ball. We also prove that $\mathbb{R}$ generates
this variety. From this, we exhibit a concrete model of the free Dedekind
$\sigma$-complete truncated Riesz spaces.
Analogous results are obtained for operations that preserve $p$-integrability
over finite measure spaces: the corresponding variety is shown to coincide with
the much studied category of Dedekind $\sigma$-complete Riesz spaces with weak
unit, $\mathbb{R}$ is proved to generate this variety, and a concrete model of
the free Dedekind $\sigma$-complete Riesz spaces with weak unit is exhibited.Comment: Changed the definition of "conditionally partitionable measure
space", results unchanged; minor change

### Barr-Exact Categories and Soft Sheaf Representations

It has long been known that a key ingredient for a sheaf representation of a
universal algebra A consists in a distributive lattice of commuting congruences
on A. The sheaf representations of universal algebras (over stably compact
spaces) that arise in this manner have been recently characterised by Gehrke
and van Gool (J. Pure Appl. Algebra, 2018), who identified the central role of
the notion of softness.
In this paper, we extend the scope of this theory by replacing varieties of
algebras with Barr-exact categories, thus encompassing a number of
"non-algebraic" examples. Our approach is based on the notion of K-sheaf:
intuitively, whereas sheaves are defined on open subsets, K-sheaves are defined
on compact ones. Throughout, we consider sheaves on complete lattices rather
than spaces; this allows us to obtain point-free versions of sheaf
representations whereby spaces are replaced with frames.
These results are used to construct sheaf representations for the dual of the
category of compact ordered spaces, and to recover Banaschewski and Vermeulen's
point-free sheaf representation of commutative Gelfand rings (Quaest. Math.,
2011).Comment: 39 pages. v3: presentation improved. Title modified to reflect change
in presentatio

### On the axiomatisability of the dual of compact ordered spaces

We provide a direct and elementary proof of the fact that the category of
Nachbin's compact ordered spaces is dually equivalent to an Aleph_1-ary variety
of algebras. Further, we show that Aleph_1 is a sharp bound: compact ordered
spaces are not dually equivalent to any SP-class of finitary algebras.Comment: 10 pages. v3: minor changes. To appear in Applied Categorical
Structure

### Stone-Gelfand duality for metrically complete lattice-ordered groups

We extend Yosida's 1941 version of Stone-Gelfand duality to metrically
complete unital lattice-ordered groups that are no longer required to be real
vector spaces. This calls for a generalised notion of compact Hausdorff space
whose points carry an arithmetic character to be preserved by continuous maps.
The arithmetic character of a point is (the complete isomorphism invariant of)
a metrically complete additive subgroup of the real numbers containing $1$,
namely, either $\frac{1}{n}\mathbb{Z}$ for an integer $n = 1, 2, \dots$, or the
whole of $\mathbb{R}$. The main result needed to establish the extended duality
theorem is a substantial generalisation of Urysohn's Lemma to such "arithmetic"
compact Hausdorff spaces. The original duality is obtained by considering the
full subcategory of spaces whose each point is assigned the entire group of
real numbers. In the introduction we indicate motivations from and connections
with the theory of dimension groups.Comment: 24 pages, 2 figure

### Ideal and MacNeille completions of subordination algebras

$\mathsf{S5}$-subordination algebras were recently introduced as a
generalization of de Vries algebras, and it was proved that the category
$\mathsf{SubS5^S}$ of $\mathsf{S5}$-subordination algebras and compatible
subordination relations between them is equivalent to the category of compact
Hausdorff spaces and closed relations. We generalize MacNeille completions of
boolean algebras to the setting of $\mathsf{S5}$-subordination algebras, and
utilize the relational nature of the morphisms in $\mathsf{SubS5^S}$ to prove
that the MacNeille completion functor establishes an equivalence between
$\mathsf{SubS5^S}$ and its full subcategory consisting of de Vries algebras. We
also generalize ideal completions of boolean algebras to the setting of
$\mathsf{S5}$-subordination algebras and prove that the ideal completion
functor establishes a dual equivalence between $\mathsf{SubS5^S}$ and the
category of compact regular frames and preframe homomorphisms. Our results are
choice-free and provide further insight into Stone-like dualities for compact
Hausdorff spaces with various morphisms between them. In particular, we show
how they restrict to the wide subcategories of $\mathsf{SubS5^S}$ corresponding
to continuous relations and continuous functions between compact Hausdorff
spaces

### Vietoris endofunctor for closed relations and its de Vries dual

We generalize the classic Vietoris endofunctor to the category of compact
Hausdorff spaces and closed relations. The lift of a closed relation is done by
generalizing the construction of the Egli-Milner order. We describe the dual
endofunctor on the category of de Vries algebras and subordinations. This is
done in several steps, by first generalizing the construction of Venema and
Vosmaer to the category of boolean algebras and subordinations, then lifting it
up to $\mathsf{S5}$-subordination algebras, and finally using MacNeille
completions to further lift it to de Vries algebras. Among other things, this
yields a generalization of Johnstone's pointfree construction of the Vietoris
endofunctor to the category of compact regular frames and preframe
homomorphisms

### A generalization of de Vries duality to closed relations between compact Hausdorff spaces

Stone duality generalizes to an equivalence between the categories StoneR of Stone spaces and closed relations and BAS of boolean algebras and subordination relations. Splitting equivalences in StoneR yields a category that is equivalent to the category KHausR of compact Hausdorff spaces and closed relations. Similarly, splitting equivalences in BAS yields a category that is equivalent to the category De VS of de Vries algebras and compatible subordination relations. Applying the machinery of allegories then yields that KHausR is equivalent to De VS, thus resolving a problem recently raised in the literature.The equivalence between KHausR and De VS further restricts to an equivalence between the category KHausR of compact Hausdorff spaces and continuous functions and the wide subcategory De VF of De VS whose morphisms satisfy additional conditions. This yields an alternative to de Vries duality. One advantage of this approach is that composition of morphisms is usual relation composition

### A Finite Axiomatization of Positive MV-Algebras

Positive MV-algebras are the subreducts of MV-algebras with respect to the signature {âŠ•,âŠ™,âˆ¨,âˆ§,0,1}. We provide a finite quasi-equational axiomatization for the class of such algebras