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1

A Discrete Representation for Dicomplemented Lattices

Düntsch I., Kwuida L., Orłowska E.

Fundamenta Informaticae

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2017

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Vol. 156, nr 3/4

281--295

EN

Dicomplemented lattices were introduced as an abstraction of Wille’s concept algebras which provided negations to a concept lattice. We prove a discrete representation theorem for the class of dicomplemented lattices. The theorem is based on a topology free version of Urquhart’s representation of general lattices.

2

Algebras of Definable and Rough Sets in Quasi Order-based Approximation Spaces

Kumar A., Banerjee M.

Fundamenta Informaticae

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2015

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Vol. 141, nr 1

37--55

EN

A pair of approximation operators, based on the notion of granules in generalized approximation spaces, was studied in an earlier work by the authors. In this article, we investigate algebraic structures formed by the definable sets and also by the rough sets determined by this pair of approximation operators. The definable sets are open sets of an Alexandrov topological space, and form a completely distributive lattice in which the set of completely join irreducible elements is join dense. The collection of rough sets also forms a similar structure. Representation results for such classes of completely distributive lattices as well as Heyting algebras in terms of definable and rough sets are obtained. Further, two unary operators on rough sets are considered, making the latter constitute a structure that is named a ‘rough lattice’. Representation results for rough lattices are proved.

3

Quantitative Analysis of Lattice-valued Kripke Structures

Pan H., Zhang M., Wu H., Chen Y

Fundamenta Informaticae

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2014

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Vol. 135, nr 3

269--293

EN

To model and analyze systems with multi-valued information, in this paper, we present an extension of Kripke structures in the framework of complete residuted lattices, which we will refer to as lattice-valued Kripke structures (LKSs). We then show how the traditional trace containment and equivalence relations, can be lifted to the lattice-valued setting, and we introduce two families of lattice-valued versions of the relations. Further, we explore some interesting properties of these relations. Finally, we provide logical characterizations of our relations by a natural extension of linear temporal logic.

4

On k-cyclic SHn-algebra

Fernandez A.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2011

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Vol. 59, nr 3

303-304

EN

In this work we consider a new class of algebra called k-cyclic SHn-algebra (A, T) where A is an SHn-algebra and T is a lattice endomorphism such that Tk(x) = x, for all x, k is a positive integer. The main goal of this paper is to show a Priestley duality theorem for k-cyclic SHn-algebra.

5

Discrete Dualities for Heyting Algebras with Operators

Orłowska E., Rewitzky I.

Fundamenta Informaticae

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2007

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Vol. 81, nr 1-3

275-295

EN

Discrete dualities are presented for Heyting algebras with various modal operators, for Heyting algebras with an external negation, for symmetric Heyting algebras, and for Heyting-Brouwer algebras.