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1

Morphisms in Binary Rough Set Theory

Zheng Tingting

Fundamenta Informaticae

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2020

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Vol. 172, nr 4

413--434

EN

Some scholars have proposed homomorphisms between information systems based on consistent functions. However, the binary relations in the codomain induced by consistent functions thoroughly depend on the binary relations induced by the original domain systems. This paper introduces the concept of core knowledge to analyze the intrinsical topology structures of binary approximation spaces, binary knowledge bases and binary information systems. Because of the three different categories, we use the term "morphism" from category theory to depict the communication into the three categories. A morphism can be regarded as the composition of a natural projection induced by core knowledge and an embedding, which are more general than homomorphisms. What's more, this paper proposes the notion of isomorphism and shows that the two isomorphic categories can be seen as one category based on the topological invariance. Considering that the reduction of knowledge and attributes should be based on the premise of maintaining the structure of core knowledge, isomorphisms will provide the theoretical basis of the reduction.

2

co-Semi-analytic Functors

Zawadowski Marek

Fundamenta Informaticae

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2019

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Vol. 164, nr 4

359--373

EN

We give an abstract characterization of the category of co-semi-analytic functors and describe an action of semi-analytic functors on co-semi-analytic functors.

3

More on Square-free Words Obtained from Prefixes by Permutations

Ochem P.

Fundamenta Informaticae

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2014

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Vol. 132, nr 1

109--112

EN

An infinite square-free word w over the alphabet Σ3 = {0, 1, 2} is said to have a k-stem σ if |σ| = k and w = σw1w2· · · where for each i, there exists a permutation πi of Σ3 which extended to a morphism gives wi= πi (σ). Harju proved that there exists an infinite k-stem word for k = 1, 2, 3, 9 and 13 ≤ k ≤ 19, but not for 4 ≤ k ≤ 8 and 10 ≤ k ≤ 12. He asked whether k-stem words exist for each k ≥ 20. We give a positive answer to this question. Currie has found another construction that answers Harju’s question.

4

Row and Column Spaces of Matrices over Residuated Lattices

Belohlavek R., Konecny J.

Fundamenta Informaticae

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2012

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Vol. 115, nr 4

279-295

EN

We present results regarding row and column spaces of matrices whose entries are elements of residuated lattices. In particular, we define the notions of a row and column space for matrices over residuated lattices, provide connections to concept lattices and other structures associated to such matrices, and show several properties of the row and column spaces, including properties that relate the row and column spaces to Schein ranks of matrices over residuated lattices. Among the properties is a characterization of matrices whose row (column) spaces are isomorphic. In addition, we present observations on the relationships between results established in Boolean matrix theory on one hand and formal concept analysis on the other hand.

5

On effective non-ample divisors

Jelonek Z.

Demonstratio Mathematica

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2007

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Vol. 40, nr 1

33-35

EN

Let X be a smooth complete algebraic variety. Let f : X approaches Y be a morphism from X to another algebraic variety Y, which is neither finite nor constant. Then X admits an eective non-ample divisor. In particular, if X is a smooth complete variety with Picard number one, then every non-constant morphism f : X approaches Y is finite and the variety f(X) is projective.

6

A note on homomorphisms of inner product spaces

Pulmannova S.

Demonstratio Mathematica

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2005

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Vol. 38, nr 4

997--1004

EN

Recently, Buhagiar and Chetcuti [1] have shown that if V1 and V2 are two separable, real inner product spaces such that the modular ortholattices of their finite and cofmite subspaces are algebraically isomorphic, then V1 and V2 are isomorphic as inner product spaces. Their proof is based on the properties of inner product spaces, in particular it makes use of Gleason's theorem. In this note we show, using techniques of projective geometry, that their result holds for any inner product spaces, real, complex or quaternionic, of dimension at least three, not necessarily separable. We also consider the case when the algebraic isomorphism is replaced by a homomorphism, and the case when the underlying fields of V1 and V2 are not the same.

7

The synthesis problem for Elementary Net Systems with Inhibitor Arcs

Pietkiewicz-Koutny M.

Fundamenta Informaticae

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1999

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Vol. 40, Nr 2,3

251-283

EN

We investigate the synthesis problem for the Elementary Net Systems with Inhibitor Arcs (ENI-systems) executed according to the a-priori semantics. We characterise transition systems generated by ENI-systems, called TSENI transition systems, by adapting the notion of a step transition system whose arcs are labelled by sets of concurrently executed events. The relationship between the ENI-systems and TSENI transition systems is established via the notion of a region. We define, and show consistency of, two behaviour preserving translations between nets and transition systems. We also discuss how to optimise the synthesis procedure by using only minimal regions and selected inhibitor arcs.

8

On the decidability of some problems concerning morphisms, regular sets, and post correspondence problem

Turakainen P.

Fundamenta Informaticae

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1999

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Vol. 38, Nr 1,2

209-221

EN

Regular languages are divided into equivalence classes according to the lengths of the words and both the universal and the existential equivalence of rational transductions on the set of these classes is studied. It is shown that the cardinality equivalence problem is undecidable for e-free finite substitutions. The morphic replication equivalence problem is arithmetized and an application to word equations is presented. Finally, the generalized Post correspondence problem is modified by using a single inverse morphism or a single finite substitution or its inverse instead of two morphisms.

9

Regular morphisms on semicommutations

Ochmański E.

Prace Instytutu Podstaw Informatyki Polskiej Akademii Nauk

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1998

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nr 850

3-16

EN

Each semicommutation (A, q) determines the class REG (A) - the family of all q-closed and regular subsets of A*. Let (A, q) and B, g) be two semicommutations. A morphism f:A*->B* is said to be regular if and only if f (f(L)) REG (B) whenever L REG (A). A characterization of regular semicommutations (the case A=B and f=id) was given in [OW93]. In this paper we prove regularity of some special class of morphism - path-preserving morphisms. Using this result and the former characterization of regular semicommutations we obtain a criterion of regularity for the class of injective morphisms. Finally, we prove quite weak sufficient condition of regularity for arbitrary morphisms. We do not know, if the condition is also a necessary one.

PL

Każda półprzemienność (A, q) wyznacza klasę REG (A) - rodzinę wszystkich q-domkniętych i regularnych podzbiorów A*. Jeśli (A, q) i (B, g) są dwoma półprzemiennościami, to morfizm f:A*->B* nazywamy regularnym wtedy i tylko wtedy, gdy dla każdego L REG (A) domknięcie jego obrazu f (f(L)) należy do REG (B). Przypadek regularnych półprzemienności (A=B i f=id) został scharakteryzowany w [OW93]. W tej pracy pokazujemy regularność pewnej szczególnej klasy morfizmów, mianowicie morfizmów zachowujących ścieżki zależności. Wynik ten, w połączeniu ze wzmiankowaną wyżej charakteryzacją regularnych półprzemienności, pozwala sformułować i udowodnić kryterium regularności dla morfizmów różnowartościowych. Na zakończenie pokazujemy pewien warunek wystarczający dla regularności morfizmów. Pozostaje pytaniem otwartym, czy warunek ten jest również konieczny.