A discrete Sugeno integral on a bounded distributive lattice L is defined as an idempotent weighted polynomial. Another possibility for axiomatization of integrals to consider compatible aggregation functions, uniquely extending the L-valued fuzzy measures. This paper aims study mentioned unique extension property concerning possible non-distributive lattices. We show that this equivalent distr...

‎We give a simple and independent axiomatization of reticulations on residuated lattices‎, ‎which were axiomatized by five conditions in [C‎. ‎Mureşan‎, ‎The reticulation of a residuated lattice‎, ‎Bull‎. ‎Math‎. ‎Soc‎. ‎Sci‎. ‎Math‎. ‎Roumanie‎ ‎51 (2008)‎, ‎no‎. ‎1‎, ‎47--65]‎. ‎Moreover‎, ‎we show that reticulations can be considered as lattice homomorphisms between residuated lattices and b...

In this work we develop a categorical duality for certain classes of manysorted algebras, called many-sorted lattices because each sort admits a structure of distributive lattice. This duality is strongly based on the Priestley duality for distributive lattices developed in [3] and [4] and on the representation of many sorted lattices with operators given by Sofronie-Stokkermans in [6]. In this...

The concepts of P-compactness, countable P-compactness, the P-Lindelöf property are introduced in L-topological spaces by means of preopen L-sets and their inequalities when L is a complete DeMorgan algebra. These definitions do not rely on the structure of the basis lattice L and no distributivity in L is required. They can also be characterized by means of preclosed L-sets and their inequalit...

In this paper, we first introduce the concept of single elements in a module. A systematic study of single elements in the AlgL-module U is initiated, where L is a completely distributive subspace lattice on a Hilbert space H. Furthermore, as an application of single elements, we study module isomorphisms between norm closed AlgN -modules, where N is a nest, and obtain the following result: Sup...

We consider the problem of interpolating functions partially defined over a distributive lattice, by means of lattice polynomial functions. Goodstein’s theorem solves a particular instance of this interpolation problem on a distributive lattice L with least and greatest elements 0 and 1, resp.: Given a function f : {0, 1}n → L, there exists a lattice polynomial function p : Ln → L such that p|{...

For a finite lattice L, let EL denote the reflexive and transitive closure of the join-dependency relation on L, defined on the set J(L) of all join-irreducible elements of L. We characterize the relations of the form EL, as follows: Theorem. Let E be a quasi-ordering on a finite set P . Then the following conditions are equivalent: (i) There exists a finite lattice L such that 〈J(L),EL〉 is iso...

Some well-known algebras of logic or other algebraic models of non-classical reasoning, such as MV-algebras or orthomodular lattices, can be regarded as lattices with antitone involutions, i.e., bounded lattices with the property that all principal filters (or ideals) are equipped with antitone involutions. This simple observation is behind the so-called “basic algebras” that were introduced in...

The problem of imbedding an abstract distributive lattice in a Boolean algebra by an algebraic extension was suggested to the writer by M. H. Stone in 1933. Hausdorfff had already given a solution of this problem for the case where the given distributive lattice was a ring of point sets. A solution for the abstract case was presented by the writer to the Harvard Mathematical Colloquium (1934), ...