We continue our research plan of developing the theory small and locally spaces, proposing this as a realisation Grothendieck’s idea tame topology on level general topology. In paper, we develop Heyting spaces prove new version Esakia Duality for such spaces. To do this, notice that spectral may be seen sober with all smops compact introduce method standard spectralification. This helps to unde...

The Semi Heyting Almost Distributive Lattice (SHADL) is a mathematical framework that combines the concepts of semi algebra and almost distributive lattice. This abstract highlights applications SHADL in various domains

It is a tempting idea to use formal existence proofs as a means to precisely and verifiably express algorithmic ideas. This is clearly possible for “constructive” proofs, which are informally understood via the BrouwerHeyting-Kolmogorov interpretation (BHK-interpretation for short). This interpretation of intuitionistic (and minimal) logic explains what it means to prove a logically compound st...

We begin by recalling the general theory of adjoints on finite semilattices. A finite join semilattice with 0 is a lattice, with the naturally induced meet operation. Thus a finite lattice S can be regarded as a semilattice in two ways, either S = 〈S,+, 0〉 or S = 〈S,∧, 1〉. Given a (+, 0)-homomorphism g : S → T , define the adjoint h : T → S by h(t) = ∑ {s ∈ S : gs ≤ t} so that gs ≤ t iff s ≤ ht...

We describe the semilattice of ordered compactifications ofX×Y smaller than βoX×βoY whereX and Y are certain totally ordered topological spaces, and where βoZ denotes the Stone–Čech orderedor Nachbin-compactification of Z. These basic cases are used to illustrate techniques for describing the semilattice of ordered compactifications ofX×Y smaller than βoX×βoY for arbitrary totally ordered topol...

We give a new order-theoretic characterization of a complete Heyting and co-Heyting algebra C. This result provides an unexpected relationship with the field of Nash equilibria, being based on the so-called Veinott ordering relation on subcomplete sublattices of C, which is crucially used in Topkis’ theorem for studying the order-theoretic stucture of Nash equilibria of supermodular games. Intr...