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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

Properties of differences in B-rings

Dadej A., Halik K.

Journal of Mathematics and Applications

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2011

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Vol. 34

5-13

EN

Motivated by Pettis' extensions of Sierpinski theorems on generated families of sets, we consider B-rings, a generalization of the notion of Boolean algebras, and present their various properties. In particular, we discuss properties of differences which will be used in the proofs of results given in our forthcoming papers.

3

Algorithm and program for finding minimal and quasi-minimal cuts in graphs

Grishkeyich A., Piątek Ł.

Zeszyty Naukowe Wydziału Elektrotechniki i Automatyki Politechniki Gdańskiej

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2008

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Nr 25

49-52

EN

It was found, that the set of minimum cuts, separating two chosen vertices in graph, have the structure of distributive lattice. It was developed an effective procedure for finding the set of 1, 2 and 3 elements cuts in graph based on the consideration of distributive lattice of the set of minimum cuts. The procedure consists of first, the algorithm for finding indecomposable minimal cuts of distributive lattice. Second, algorithm for synthesis, using resulting subset from stage one, of the entire set of minimum cuts. The third, is the algorithm for describing the set of quasi-minimum (close to the minimum, next to minimum) cuts in the form of sum of distributive lattices of minimal cuts found for the modified function of weight. The computer program, implementing these algorithms, is presented with examples.

PL

Ustalono, że zbiór minimalnych przekrojów, rozdzielających dwa zadane wierzchołki grafu, z wprowadzonymi na i operacjami ma strukturę kraty dystrybutywnej. Opracowano skuteczną algorytmiczną procedurę znajdowania zbioru jed dwu i trzy elementowych przekrojów grafu bazującą na rozpatrzeniu dystrybutywnych krat zbioru minimalnych przekrojów grafu. Procedura składa się z, po pierwsze, algorytmu szukania nierozkładalnych minimalnych przekrojów Dystrybutywnej, po drugie, algorytmu syntezy po tym podzbiorze w kracie dystrybutywnej całego poszukiwanego zbioru minimalnych przekrojów i, po trzecie, algorytmu opisu zbioru quasi-minimalnych (bliskich do minimalnych, następnych po minimalnych) przekrojów w formie sumy krat dystrybutywnych minimalnych przekrojów, znalezionych dla zmodyfikowanej funkcji wagi. Przedstawiono zrealizowany program komputerowy i przedstawiono przykłady pracy programu.

4

Multi-Valued MSO Logics Over Words and Trees

Droste M., Kuich W., Rahonis G.

Fundamenta Informaticae

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2008

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Vol. 84, nr 3-4

305-327

EN

We introduce multi-valued Büchi and Muller automata over distributive lattices and a multi-valued MSO logic for infinite words. For this logic, we prove the expressive equivalence of w-recognizable and MSO-definable infinitary formal power series over distributive lattices with negation function. Then we consider multi-valued Muller tree automata and a multi-valued MSO logic for trees over distributive lattices. For this logic, we establish a version of Rabin's theorem for infinitary tree series.

5

An Algebraic Foundation for Linguistic Reasoning

Huynh. V.N., Murai T., Nakamori Y.

Fundamenta Informaticae

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2007

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Vol. 78, nr 2

271-294

EN

It is well known that algebraization has been successfully applied to classical and non-classical logics (Rasiowa and Sikorski, 1968). Following this direction, an ordered-based approach to the problem of finding out a tool to describe algebraic semantics of Zadeh's fuzzy logic has been introduced and developed by Nguyen Cat-Ho and colleagues during the last decades. In this line of research, RH_algebra has been introduced in [20] as a unified algebraic approach to the natural structure of linguistic domains of linguistic variables. It was shown that every RH_algebra of a linguistic variable with a chain of the primary terms is a distributive lattice. In this paper we will examine algebraic structures of RH_algebras corresponding to linguistic domains having exactly two distinct primary terms, one being an antonym of the other, called symmetrical RH_algebras. Computational results for the relatively pseudo-complement operation in these algebras will be given.

6

Algorytm przeliczania dwuelementowych minimalnych przekrojów grafa skierowanego

Grishkevich A.A.

Informatyka Teoretyczna i Stosowana

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2006

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R. 6, nr 10

85-100

PL

Zaproponowano sposób kodowania kraty dystrybutywnej minimalnych przekrojów grafu. Opracowano algorytm przeliczania minimalnych dwuelementowych przekrojów grafu w kracie dystrybutywnej minimalnych przekrojów. Zaprezentowano przykład działania algorytmu.

EN

Coding of the distributive lattice of the minimal cuts graphs is offered. Enumerating algorithm of the minimal two-element cuts of the graph in a distributive lattice of the minimal cuts is developed. The example of work of the algorithm is presented.

7

Distributive lattice of minimum cut sets of a directed graph

Grishkevich A.A.

Informatyka Teoretyczna i Stosowana

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2004

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R. 4, nr 7

7-22

EN

The algebraic operations above cut sets of a digraph are entered. Is demonstrated, that the set of minimum cut sets separating two vertices of the graph, bas structure of a distributive lattice. The concept of a undecomposable (nonreducible) cut set is reviewed and the representation of set of minimum cut sets is obtained on the basis of a subset of undecomposable cut sets. The special collections of cut sets are established, in which one the nonreducible cut set is extreme. The obtained results are illustrated by example.

8

Left-outtermost extension of some varieties

Płonka J.

Demonstratio Mathematica

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2003

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Vol. 36, nr 1

37--51

EN

Let r : F - > N be a type of algebras F is a nonempty set of fundamental operation symbols and N is the set of all positive integers. An identity ip fa if) of type T we call left-outermost if the left-outermost variables in ip and ip are the same. For a variety V of type r we denote by Vi the variety of type r defined by all left-outermost identities from Id(V). Vl is called the left-outermost extension of V. In this paper we study minimal generics, subdirectly irreducible algebras and lattices of subvarieties in left-outermost extensions of some generalizations of the variety D of all distributive lattices.