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On the lattice of tolerances for a finite chain

Górnicka A., Grygiel J., Tyrala I.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

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2016

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Vol. 21

25--30

EN

We provide some description of the lattice of tolerances for a finite chain, pointing to the skeleton tolerance as a special element of this lattice. In particular, we prove that the lattice of all glued tolerances of an n-element chain is isomorphic to the lattice of all tolerances of an n- 1-element chain nad at the same time is a principal filter of the lattice of an n-element chain.

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Congruences of Edge-bipartite Graphs with Applications to Grothendieck Group Recognition. [Part 1], Inflation Algorithm Revisited

Mróz A.

Fundamenta Informaticae

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2016

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Vol. 146, nr 2

121--144

EN

We study edge-bipartite graphs (bigraphs), a class of signed graphs, by means of the inflation algorithm which relies on performing certain elementary transformations on a given bigraph Δ, or equivalently, on the associated integral quadratic form qΔ: Zn → Z, preserving Gram Z-congruence. The ideas are inspired by classical results of Ovsienko and recent studies of Simson started in [SIAM J. Discr. Math. 27 (2013), 827-854], concerning classifications of integral quadratic and bilinear forms, and their Coxeter spectral analysis. We provide few modifications of the inflation algorithm and new estimations of its complexity for positive and principal loop-free bigraphs. We discuss in a systematic way the behavior and computational aspects of inflation techniques. As one of the consequences we obtain relatively simple proofs of several interesting properties of quadratic forms and their roots, extending known facts. On the other hand, the results are a first step of a solution of a variant of Grothendieck group recognition, a difficult combinatorial problem arising in representation theory of finite dimensional algebras and their derived categories, which we discuss in Part II of this two parts article with the same main title.

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Bisimulation Cuts For Structuring Markov Transition Systems

Doberkat E.-E.

Fundamenta Informaticae

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2016

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Vol. 149, nr 4

363--383

EN

In Universal Algebra the structure of congruences for algebraic systems is fairly well investigated, and the relationship to the structure of the underlying system proper is well known. We propose a first step into this direction for studying the structure of congruences for stochastic relations. A Galois connection to a certain class of Boolean σ-algebras is exploited, atoms and antiatoms are identified, and it is show that a σ-basis exists. These constructions are applied to the problem of finding bisimulation cuts of a congruence. It cuts the relation through a span of morphisms with a minimum of joint events.

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On the Accuracy of Rough Approximations of Regular Languages

Torres G. M.

Fundamenta Informaticae

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2014

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Vol. 132, nr 4

533--545

EN

In this paper we attempt to measure the accuracy of approximations of regular languages by languages in +−varieties (as defined by Eilenberg). These approximations are upper approximations in the sense of Pawlak’s rough set theory with respect to congruences belonging to the variety of congruences corresponding to the given +−variety. In our approach, the accuracy of an approximation is measured by the relative density of the object language in the approximation language and the asymptotic behavior of this quotient. In particular, we apply our measures of accuracy to k-definite, reverse k-definite, i, j-definite and k-testable approximations.

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Numerical investigation of least prime primitive roots modulo a prime number

Paszkiewicz A.

Biuletyn Wojskowej Akademii Technicznej

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1999

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Vol. 48, nr 10

41-52

EN

For each of the 105,097,564 primes p from the interval [3,2 31] there exists a primitive root h(p)

PL

Dla każdej spośród 105,007,564 liczb pierwszych p z przedziału [3,2 31] istnieje pierwiastek pierwotny h(p)