Wehrung, Friedrich

It is well known that the real spectrum of any commutative unital ring, and the ℓ-spectrum of any Abelian lattice-ordered group with order-unit, are all completely normal spectral spaces. We prove the following results: (1) Every real spectrum can be embedded, as a spectral subspace, into some ℓ-spectrum. (2) Not every real spectrum is an ℓ-spectru...

Wehrung, Friedrich

In an earlier paper we established that every second countable, completely normal spectral space is homeomorphic to the ℓ-spectrum of some Abelian ℓ-group. We extend that result to ℓ-spectra of vector lattices over any countable totally ordered division ring k. Extending our original machinery, about finite lattices of polyhedra, from linear to aff...

Gehrke, Mai Jakl, Tomáš Reggio, Luca
Published in
Foundations of Software Science and Computation Structures

A systematic theory of structural limits for finite models has been developed by Nešetřil and Ossona de Mendez. It is based on the insight that the collection of finite structures can be embedded, via a map they call the Stone pairing , in a space of measures, where the desired limits can be computed. We show that a closely related but finer graine...

Bezhanishvili, Nick Holliday, Wesley Halcrow

The standard topological representation of a Boolean algebra via the clopen sets of a Stone space requires a nonconstructive choice principle, equivalent to the Boolean Prime Ideal Theorem. In this paper, we describe a choice-free topological representation of Boolean algebras. This representation uses a subclass of the spectral spaces that Stone u...

Reggio, Luca

The unifying theme of the thesis is the semantic meaning of logical quantifiers. In their basic form quantifiers allow to state theexistence, or non-existence, of individuals satisfying a property. As such, they encode the richness and the complexity of predicate logic, as opposed to propositional logic. We contribute to the semantic understanding ...

Mundici, Daniele
Published in
Soft Computing

de Finetti’s Dutch book theorem explains why probability has to be additive on disjunctions of incompatible yes–no (Boolean) events. The theorem holds verbatim also for continuous events, or random variables, as formalized in Łukasiewicz logic. Independence, a subtler notion than incompatibility, has a probability-free definition: two events are lo...

Gehrke, Mai Van Gool, Sam

It has long been known in universal algebra that any distributive sublattice of congruences of an algebra which consists entirely of commuting congruences yields a sheaf representation of the algebra. In this paper we provide a gener-alisation of this fact and prove a converse of the generalisation. To be precise, we exhibit a one-to-one correspond...

Wehrung, Friedrich

The problem of determining the range of a given functor arises in various parts of mathematics.We present a sample of such problems, with focus on various functors, arising in the contexts of nonstable K_0-theory of rings, congruence lattices of universal algebras, spectral spaces of ring-like objects.We also sketch some of the ideas involved in th...

Lawson, Mark V.
Published in
Semigroup Forum

This paper continues the study of a class of inverse monoids, called Tarski monoids, that can be regarded as non-commutative generalizations of the unique countable, atomless Boolean algebra. These inverse monoids are related to a class of étale topological groupoids under a non-commutative generalization of classical Stone duality and, significant...

Bezhanishvili, Guram
Published in
Algebra universalis

There is a well-known correspondence between Heyting algebras and S4-algebras. Our aim is to extend this correspondence to distributive lattices by defining analogues of S4-algebras for them. For this purpose, we introduce binary relations on Boolean algebras that resemble de Vries proximities. We term such binary relations lattice subordinations. ...