45 research outputs found

### Stone duality above dimension zero: Axiomatising the algebraic theory of C(X)

It has been known since the work of Duskin and Pelletier four decades ago
that KH^op, the category opposite to compact Hausdorff spaces and continuous
maps, is monadic over the category of sets. It follows that KH^op is equivalent
to a possibly infinitary variety of algebras V in the sense of Slominski and
Linton. Isbell showed in 1982 that the Lawvere-Linton algebraic theory of V can
be generated using a finite number of finitary operations, together with a
single operation of countably infinite arity. In 1983, Banaschewski and Rosicky
independently proved a conjecture of Bankston, establishing a strong negative
result on the axiomatisability of KH^op. In particular, V is not a finitary
variety--Isbell's result is best possible. The problem of axiomatising V by
equations has remained open. Using the theory of Chang's MV-algebras as a key
tool, along with Isbell's fundamental insight on the semantic nature of the
infinitary operation, we provide a finite axiomatisation of V.Comment: 26 pages. Presentation improve

### Quantifiers on languages and codensity monads

This paper contributes to the techniques of topo-algebraic recognition for
languages beyond the regular setting as they relate to logic on words. In
particular, we provide a general construction on recognisers corresponding to
adding one layer of various kinds of quantifiers and prove a corresponding
Reutenauer-type theorem. Our main tools are codensity monads and duality
theory. Our construction hinges on a measure-theoretic characterisation of the
profinite monad of the free S-semimodule monad for finite and commutative
semirings S, which generalises our earlier insight that the Vietoris monad on
Boolean spaces is the codensity monad of the finite powerset functor.Comment: 30 pages. Presentation improved and details of several proofs added.
The main results are unchange

### A model category for modal logic

We define Quillen model structures on a family of presheaf toposes arising
from tree unravellings of Kripke models, leading to a homotopy theory for modal
logic. Modal preservation theorems and the Hennessy-Milner property are
revisited from a homotopical perspective.Comment: 25 page

### Codensity, profiniteness and algebras of semiring-valued measures

We show that, if S is a finite semiring, then the free profinite S-semimodule
on a Boolean Stone space X is isomorphic to the algebra of all S-valued
measures on X, which are finitely additive maps from the Boolean algebra of
clopens of X to S. These algebras naturally appear in the logic approach to
formal languages as well as in idempotent analysis. Whenever S is a (pro)finite
idempotent semiring, the S-valued measures are all given uniquely by continuous
density functions. This generalises the classical representation of the
Vietoris hyperspace of a Boolean Stone space in terms of continuous functions
into the Sierpinski space.
We adopt a categorical approach to profinite algebra which is based on
profinite monads. The latter were first introduced by Adamek et al. as a
special case of the notion of codensity monads.Comment: 21 pages. Presentation improved. To appear in the Journal of Pure and
Applied Algebr

### Beth definability and the Stone-Weierstrass Theorem

The Stone-Weierstrass Theorem for compact Hausdorff spaces is a basic result
of functional analysis with far-reaching consequences. We show that this
theorem is a consequence of the Beth definability property of a certain
infinitary equational logic, stating that every implicit definition can be made
explicit.Comment: 20 pages. v2: minor changes, added a "Conclusion" sectio

### Model completions for universal classes of algebras: necessary and sufficient conditions

Necessary and sufficient conditions are presented for the (first-order)
theory of a universal class of algebraic structures (algebras) to admit a model
completion, extending a characterization provided by Wheeler. For varieties of
algebras that have equationally definable principal congruences and the compact
intersection property, these conditions yield a more elegant characterization
obtained (in a slightly more restricted setting) by Ghilardi and Zawadowski.
Moreover, it is shown that under certain further assumptions on congruence
lattices, the existence of a model completion implies that the variety has
equationally definable principal congruences. This result is then used to
provide necessary and sufficient conditions for the existence of a model
completion for theories of Hamiltonian varieties of pointed residuated
lattices, a broad family of varieties that includes lattice-ordered abelian
groups and MV-algebras. Notably, if the theory of a Hamiltonian variety of
pointed residuated lattices admits a model completion, it must have
equationally definable principal congruences. In particular, the theories of
lattice-ordered abelian groups and MV-algebras do not have a model completion,
as first proved by Glass and Pierce, and Lacava, respectively. Finally, it is
shown that certain varieties of pointed residuated lattices generated by their
linearly ordered members, including lattice-ordered abelian groups and
MV-algebras, can be extended with a binary operation in order to obtain
theories that do have a model completion.Comment: 32 page

### 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

### Arboreal Categories and Equi-resource Homomorphism Preservation Theorems

The classical homomorphism preservation theorem, due to {\L}o\'s, Lyndon and
Tarski, states that a first-order sentence $\phi$ is preserved under
homomorphisms between structures if, and only if, it is equivalent to an
existential positive sentence $\psi$. Given a notion of (syntactic) complexity
of sentences, an "equi-resource" homomorphism preservation theorem improves on
the classical result by ensuring that $\psi$ can be chosen so that its
complexity does not exceed that of $\phi$.
We describe an axiomatic approach to equi-resource homomorphism preservation
theorems based on the notion of arboreal category. This framework is then
employed to establish novel homomorphism preservation results, and improve on
known ones, for various logic fragments, including first-order, guarded and
modal logics.Comment: 44 pages. v3: expanded the Introduction, added a new Section 8,
changed the title to reflect the focus of the pape

### 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