We give several characterizations of discrete Sugeno integrals over bounded distributive lattices, as particular cases of lattice polynomial functions, that is functions which can be represented in the language of bounded lattices using variables and constants. We also consider the subclass of term functions as well as the classes of symmetric polynomial functions and weighted minimum and maxim...

Bounded residuated lattice ordered monoids (R -monoids) are a common generalization of pseudo-BL-algebras and Heyting algebras, i.e. algebras of the non-commutative basic fuzzy logic (and consequently of the basic fuzzy logic, the Łukasiewicz logic and the non-commutative Łukasiewicz logic) and the intuitionistic logic, respectively. In the paper we introduce and study classes of filters of bou...

The two main objectives of this paper are (a) to prove topological duality theorems for semilattices and bounded lattices, and (b) to show that the topological duality from (a) provides a construction of canonical extensions of bounded lattices. The paper is the first of two parts. The main objective of the sequel [MJ2] is to establish a characterization of lattice expansions, i.e., lattices wi...

The lattice of z-ideals of the ring C(X) of real-valued continuous functions on a completely regular Hausdorff space X has been shown by Mart́ınez and Zenk to be a complete Heyting algebra with certain properties. We show that these properties are due only to the fact that C(X) is an f -ring with bounded inversion. This we do by studying lattices of algebraic z-ideals of abstract f -rings with b...

We construct a complete lattice Z such that the binary supremum function sup : Z × Z → Z is discontinuous with respect to the product topology on Z × Z of the Scott topologies on each copy of Z. In addition, we show that bounded completeness of a complete lattice Z is in general not inherited by the dcpo C(X,Z) of continuous functions from X to Z where X may be any topological space and where o...

S-extremal strongly modular lattices maximize the minimum of the lattice and its shadow simultaneously. They are a direct generalization of the s-extremal unimodular lattices defined in [6]. If the minimum of the lattice is even, then the dimension of an s-extremal lattices can be bounded by the theory of modular forms. This shows that such lattices are also extremal and that there are only fin...

The uniformly frustrated layered XY model is analyzed in its Villain form. A decouple pancake vortex liquid phase is identified. It is bounded by both first-order and secondorder decoupling lines in the magnetic field versus temperature plane. These transitions, respectively, can account for the flux-lattice melting and for the flux-lattice depinning observed in the mixed phase of clean high-te...

S-extremal strongly modular lattices maximize the minimum of the lattice and its shadow simultaneously. They are a direct generalization of the s-extremal unimodular lattices defined in [6]. If the minimum of the lattice is even, then the dimension of an s-extremal lattices can be bounded by the theory of modular forms. This shows that such lattices are also extremal and that there are only fin...

A completion via Frink ideals is used to define a convex powerdomain of an arbitrary continuous lattice as a continuous lattice. The powerdomain operator is a functor in the category of continuous lattices and continuous inf-preserving maps and preserves projective limits and surjectivity of morphisms; hence one can solve domain equations in which it occurs. Analogous results hold for algebraic...

We study the problem of best approximations of a vector α ∈ R n by rational vectors of a lattice Λ ⊂ R whose common denominator is bounded. To this end we introduce successive minima for a periodic lattice structure and extend some classical results from geometry of numbers to this structure. This leads to bounds for the best approximation problem which generalize and improve former results.