Topos-theoretic semantics for modal logic usually considers structures induced by a surjective geometric morphism f : F → E . f restricts to an injective (complete) distributive lattice homomorphism ∆A : SubE(A) −→ SubF (f∗A), for each A in E and natural in A w.r.t. to pullback. Each ∆A has a right adjoint ΓA that composes with ∆A to an endofunctor ∆AΓA on the Heyting algebra SubF (f ∗A) that s...

Duality theory emerged from the work by Marshall Stone [18] on Boolean algebras and distributive lattices in the 1930s. Later in the early 1970s Larisa Maksimova [10, 11] and Hilary Priestley [15, 16] developed analogous results for Heyting algebras, topological Boolean algebras, and distributive lattices. Duality for bounded, not necessarily distributive lattices, was developed by Alstir Urquh...

Monads serve the abstract encapsulation of side effects in semantics and functional programming. Various monad-based specification languages have been introduced in order to express requirements on generic side-effecting programs. A basic role is played here by global evaluation logic, concerned with formulae which may be thought of as being universally quantified over the state space; this for...

Ordered algebras such as Boolean algebras, Heyting algebras, lattice-ordered groups, and MV-algebras have long played a decisive role in logic, although perhaps only in recent years has the significance of the relationship between the two fields begun to be fully recognized and exploited. The first aim of this survey article is to briefly trace the distinct historical roots of ordered algebras ...

The intrinsic connection between lattice theory and topology is fairly well established. For instance, the collection of open subsets of a topological subspace always forms a distributive lattice. Persistent homology has been one of the most prominent areas of research in computational topology in the past 20 years. In this paper we will introduce an alternative interpretation of persistence ba...

Simulations and bisimulations between two fuzzy automata over a complete residuated lattice were defined by Ćirić et al. (2012) [7], [8] as relations the sets of states automata. However, they act crisp relationship In particular, if there exists (forward) bisimulation automata, then languages recognized them are crisply equal. Approximate simulations introduced Stanimirović (2022) [36] aim at ...

In this paper, we show that there exist (continuum many) varieties of bi-Heyting algebras are not generated by their complete members. It follows extensions the Heyting–Brouwer logic [Formula: see text] topologically incomplete. This result provides further insight into long-standing open problem Kuznetsov yielding a negative solution reformulation from to text].

An effect in quantum mechanics on a Hilbert space H is an operator A in H such that 0 ≤ A ≤ I , and two effects, A and B, are added to get A⊕ B iff 0 ≤ A⊕ B ≤ I, where A⊕ B is the operator addition. We wish to specialize this concept to the effects that arise naturally in the formalism of quantum mechanics on phase space (Schroeck, 1996). From this, we will have an immediate generalization to H...

The techniques of natural duality theory are applied to certain finitely generated varieties of Heyting algebras to obtain optimal dualities for these varieties, and thereby to address algebraic questions about them. In particular, a complete characterisation is given of the endodualisable finite subdirectly irreducible Heyting algebras. The procedures involved rely heavily on Priestley duality...