A commutative residuated lattice is said to be subidempotent if the lower bounds of its neutral element e are idempotent (in which case they naturally constitute a Brouwerian algebra A*). It proved here that epimorphisms surjective in variety K such algebras (with or without involution), provided each finitely subdirectly irreducible B has two properties: (1) generated by e, and (2) poset prime...

We use coalgebraic methods to describe finitely generated free Heyting algebras. Heyting algebras are axiomatized by rank 0-1 axioms. In the process of constructing free Heyting algebras we first apply existing methods to weak Heyting algebras—the rank 1 reducts of Heyting algebras—and then adjust them to the mixed rank 0-1 axioms. On the negative side, our work shows that one cannot use arbitr...

In this article von Neumann’s proposal that in quantum mechanics projections can be seen as propositions is followed. However, the quantum logic derived by Birkhoff and von Neumann is rejected due to the failure of the law of distributivity. The options for constructing a distributive logic while adhering to von Neumann’s proposal are investigated. This is done by rejecting the converse of the ...

A relation on a hypergraph is a binary relation on the set consisting of the nodes and hyperedges together, and which satisfies a constraint involving the incidence structure of the hypergraph. These relations correspond to join-preserving mappings on the lattice of subhypergraphs. This paper studies the algebra of these relations, in particular the analogues of the familiar operations of compl...

A topos theoretic generalisation of the category of sets allows for modelling spaces which vary according to time intervals. Persistent homology, or more generally persistence, is a central tool in topological data analysis, which examines the structure of data through topology. The basic techniques have been extended in several different directions, encoding topological features by so called b...

Simpson introduced the lattice P of Π 1 classes under Medvedev reducibility. Questions regarding completeness in P are related to questions about measure and randomness. We present a solution to a question of Simpson about Medvedev degrees of Π 1 classes of positive measure that was independently solved by Simpson and Slaman. We then proceed to discuss connections to constructive logic. In part...

Abstract. The present paper contributes to the development of the mathematical theory of epistemic updates using the tools of duality theory. Here we focus on Probabilistic Dynamic Epistemic Logic (PDEL). We dually characterize the product update construction of PDEL-models as a certain construction transforming the complex algebras associated with the given model into the complex algebra assoc...

We describe an algebraic way to code the causal information of a discrete spacetime. The causal set C is transformed to a description in terms of the causal pasts of the events in C. This is done by an evolving set, a functor which to each event of C assigns its causal past. Evolving sets obey a Heyting algebra which is characterised by a non-standard notion of complement. Conclusions about the...

In [13] Moschovakis presents a topological model, over Baire space, for a system of intuitionistic analysis that includes relativized dependent choice, monotone bar induction (verified in [2, pp. 13-14]), Kripke’s schema, and the restriction of Brouwer’s principle for numbers to predicates without free choice-sequence parameters. Following Krol [12], one may describe this model roughly1 as foll...