The following propositions are true: (1) For every sup-semilattice L and for all elements x, y of L holds ⌈⌉L(↑x∩ ↑y) = x ⊔ y. (2) For every semilattice L and for all elements x, y of L holds ⊔ L(↓x∩↓y) = x ⊓ y. (3) Let L be a non empty relational structure and x, y be elements of L. If x is maximal in (the carrier of L) \ ↑y, then ↑x \ {x} = ↑x ∩ ↑y. (4) Let L be a non empty relational structu...

We prove that (1) for any complete lattice $L$, the set $\mathcal {D}(L)$ of all non-empty saturated compact subsets Scott space $L$ is a Heyting algebra (with reverse inclusion order); and (2) if latti

The aim of this paper is to extend the truth value table oflattice-valued convergence spaces to a more general case andthen to use it to introduce and study the quantale-valued fuzzy Scotttopology in fuzzy domain theory. Let $(L,*,varepsilon)$ be acommutative unital quantale and let $otimes$ be a binary operationon $L$ which is distributive over nonempty subsets. The quadruple$(L,*,otimes,varep...

(1) For every sup-semilattice L and for all elements x, y of L holds d−eL(↑x∩↑y) = xt y. (2) For every semilattice L and for all elements x, y of L holds ⊔ L(↓x∩↓y) = xu y. (3) Let L be a non empty relational structure and x, y be elements of L. If x is maximal in (the carrier of L)\↑y, then ↑x\{x}= ↑x∩↑y. (4) Let L be a non empty relational structure and x, y be elements of L. If x is minimal ...

We construct a version of rational Symplectic Field Theory for pairs (X, L), where X is an exact symplectic manifold, where L ⊂ X is an exact Lagrangian submanifold with components subdivided into k subsets, and where both X and L have cylindrical ends. The theory associates to (X, L) a Z-graded chain complex of vector spaces over Z2, filtered with k filtration levels. The corresponding k-level...

Given a linear equation L, a set A ⊆ [n] is L-free if A does not contain any ‘non-trivial’ solutions to L. In this paper we consider the following three general questions: (i) What is the size of the largest L-free subset of [n]? (ii) How many L-free subsets of [n] are there? (iii) How many maximal L-free subsets of [n] are there? We completely resolve (i) in the case when L is the equation px ...

The present paper has been an attempt to investigate the general fuzzy automata on the basis of complete residuated lattice-valued ($L$-GFAs). The study has been chiefly inspired from the work by Mockor cite{15, 16, 17}. Regarding this, the categorical issue of $L$-GFAs has been studied in more details. The main issues addressed in this research include: (1) investigating the relationship betwe...