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...

In this article, semi-ring and semi-algebra of sets are formalized so as to construct a measure of a given set in the next step. Although a semi-ring of sets has already been formalized in [12], that is, strictly speaking, a definition of a quasi semi-ring of sets suggested in the last few decades [14]. We adopt a classical definition of a semi-ring of sets here to avoid such a confusion. Ring ...

We describe representations of certain superconformal algebras in the semi-infinite Weil complex related to the loop algebra of a complex finite-dimensional Lie algebra and in the semi-infinite cohomology. We show that in the case where the Lie algebra is endowed with a nondegenerate invariant symmetric bilinear form, the relative semi-infinite cohomology of the loop algebra has a structure, wh...

In this paper, Kolmogorov-Sinai entropy is studied using mathematical modeling of an observer $ Theta $. The relative entropy of a sub-$ sigma_Theta $-algebra having finite atoms is defined and then   the ergodic properties of relative  semi-dynamical systems are investigated.  Also,  a relative version of Kolmogorov-Sinai theorem  is given. Finally, it is proved  that the relative entropy of a...

We define a semi-Hopf algebra which is more general than a Hopf algebra. Then we construct the supersymmetry algebra via the adjoint action on this semi-Hopf algebra. As a result we have a supersymmetry theory with quantum gauge group, i.e., quantised enveloping algebra of a simple Lie algebra. For the example, we construct the Lagrangian N =1 and N =2 supersymmetry. ∗email : [email protected] 1