Finitely generated free Heyting algebras via Birkhoff duality and coalgebra

Algebras axiomatized entirely by rank 1 axioms are algebras for a functor and thus the free algebras can be obtained by a direct limit process. Dually, the final coalgebras can be obtained by an inverse limit process. In order to explore the limits of this method we look at Heyting algebras which have mixed rank 0-1 axiomatizations. We will see that Heyting algebras are special in that they are almost rank 1 axiomatized and can be handled by a slight variant of the rank 1 coalgebraic methods

Consumption of ultra-processed foods and risk of multimorbidity of cancer and cardiometabolic diseases: a multinational cohort study

Background It is currently unknown whether ultra-processed foods (UPFs) consumption is associated with a higher incidence of multimorbidity. We examined the relationship of total and subgroup consumption of UPFs with the risk of multimorbidity defined as the co-occurrence of at least two chronic diseases in an individual among first cancer at any site, cardiovascular disease, and type 2 diabetes. Methods This was a prospective cohort study including 266,666 participants (60% women) free of cancer, cardiovascular disease, and type 2 diabetes at recruitment from seven European countries in the European Prospective Investigation into Cancer and Nutrition (EPIC) study. Foods and drinks consumed over the previous 12 months were assessed at baseline by food-frequency questionnaires and classified according to their degree of processing using Nova classification. We used multistate modelling based on Cox regression to estimate cause-specific hazard ratios (HR) and their 95% confidence intervals (CI) for associations of total and subgroups of UPFs with the risk of multimorbidity of cancer and cardiometabolic diseases. Findings After a median of 11.2 years of follow-up, 4461 participants (39% women) developed multimorbidity of cancer and cardiometabolic diseases. Higher UPF consumption (per 1 standard deviation increment, ∼260 g/day without alcoholic drinks) was associated with an increased risk of multimorbidity of cancer and cardiometabolic diseases (HR: 1.09, 95% CI: 1.05, 1.12). Among UPF subgroups, associations were most notable for animal-based products (HR: 1.09, 95% CI: 1.05, 1.12), and artificially and sugar-sweetened beverages (HR: 1.09, 95% CI: 1.06, 1.12). Other subgroups such as ultra-processed breads and cereals (HR: 0.97, 95% CI: 0.94, 1.00) or plant-based alternatives (HR: 0.97, 95% CI: 0.91, 1.02) were not associated with risk. Interpretation Our findings suggest that higher consumption of UPFs increases the risk of cancer and cardiometabolic multimorbidity. Funding Austrian Academy of Sciences, Fondation de France, Cancer Research UK, World Cancer Research Fund International, and the Institut National du Cancer

Residuated bilattices

We introduce a new product bilattice con- struction that generalizes the well-known one for interlaced bilattices and others that were developed more recently, allowing to obtain a bilattice with two residuated pairs as a certain kind of power of an arbitrary residuated lattice. We prove that the class of bilattices thus obtained is a variety, give a finite axiomatization for it and characterize the congruences of its members in terms of those of their lat- tice factors. Finally, we show how to employ our product construction to define first-order definable classes of bi- lattices corresponding to any first-order definable subclass of residuated lattices

Δ1-completions of a Poset

A join-completion of a poset is a completion for which each element is obtainable as a supremum, or join, of elements from the original poset. It is well known that the join-completions of a poset are in one-to-one correspondence with the closure systems on the lattice of up-sets of the poset. A Δ1-completion of a poset is a completion for which, simultaneously, each element is obtainable as a join of meets of elements of the original poset and as a meet of joins of elements from the original poset. We show that Δ1-completions are in one-to-one correspondence with certain triples consisting of a closure system of down-sets of the poset, a closure system of up-sets of the poset, and a binary relation between these two systems. Certain Δ1-completions, which we call compact, may be described just by a collection of filters and a collection of ideals, taken as parameters. The compact Δ1-completions of a poset include its MacNeille completion and all its join- and all its meet-completions. These completions also include the canonical extension of the given poset, a completion that encodes the topological dual of the poset when it has one. Finally, we use our parametric description of Δ1-completions to compare the canonical extension to other compact Δ1-completions identifying its relative merits

Varieties of Interlaced Bilattices

The paper contains some algebraic results on several varieties of algebras having an (interlaced) bilattice reduct. Some of these algebras have already been studied in the literature (for instance bilattices with conflation, introduced by M. Fit- ting), while others arose from the algebraic study of O. Arieli and A. Avron’s bilattice logics developed in the third author’s PhD dissertation. We extend the representation theorem for bounded interlaced bilattices (proved, among others, by A. Avron) to un- bounded bilattices and prove analogous representation theorems for the other classes of bilattices considered. We use these results to establish categorical equivalences between these structures and well-known varieties of lattices