The paper is devoted to a concept of a fuzzy ordered set and partially ordered set category fuzzyfication. The aim of this work is to define a fuzzy order relation on a usual set, on an L-valued set and on an L-set. Using the fuzzy order relation we construct fuzzy POS category and study the connection between this categories and SET(L).

There are (at least) two different approaches to constructing concept lattices – one as the concept lattice of a given formal context, the other as a distributive lattice defined as the Lindenbaum algebra of a given set of (conjunctive or disjunctive) rules. It has been pointed out, first, that these two approaches are systematically related via Birkhoff’s duality theorem between (finite) order...

By a lattice we shall always mean a distributive lattice which is bounded, i.e. has both a bottom element 0 and a top element 1. Lattice homomorphisms will always be assumed to preserve 0 and 1. By a modality on a (distributive) lattice L = (L, ∧, ∨, ≤, 0, 1) is meant a map : L → L satisfying (1) 1 = 1, (2) (x ∧ y) = x ∧ y for x, y ∈ L. The pair (L, ) will be called a modalized (distributive) l...

As an application of the dual equivalence between the category of L-spatialC-objects and the category of L-sober C-M-L-spaces, it is shown in this paper that for a fixed augmented partially ordered set A, there exists a dual equivalence between the category of A-spatial augmented partially ordered sets and the category of A-sober A-valued spaces. Then, with regard to this duality, for a fixed (...

For a distributive lattice L, we consider the problem of interpolating functions f : D → L defined on a finite set D ⊆ L, by means of lattice polynomial functions of L. Two instances of this problem have already been solved. In the case when L is a distributive lattice with least and greatest elements 0 and 1, Goodstein proved that a function f : {0, 1} → L can be interpolated by a lattice poly...

Let L be the ordered set of isomorphism types of finite lattices, where the ordering is by embeddability. We study first-order definability in this ordered set. Our main result is that for every finite lattice L, the set {l, l} is definable, where l and l are the isomorphism types of L and its opposite (L turned upside down). We shall show that the only non-identity automorphism of L is the map...

In the paper, we suggest to transform a standard sequential topology of an information graph with distributive operations into its hybrid version and parallel fragments. Such transformation allows provide efficient description calculations in resource-independent form. The result depends on available hardware resource reconfigurable system provides increase special performance comparison initia...

In this paper,  it is shown that the category of $L$-ordered fuzzifying convergence spaces contains the category of pretopological $L$-ordered fuzzifying convergence spaces as a bireflective subcategory and the latter contains the category of topological $L$-ordered fuzzifying convergence spaces as a bireflective subcategory. Also, it is proved that the category of $L$-ordered fuzzifying conver...