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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

Some properties of pseudo-BCI algebras

Dymek G.

Commentationes Mathematicae

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2018

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Vol. 58, No. 1/2

19--35

EN

In this paper the notion of an essential closed deductive system of a pseudo-BCI algebra is defined and investigated. Among other things, it is proved that such a deductive system contains all coatoms of the pseudo-BCI algebra. Also, the notions of homomorphisms and semihomomorphisms of pseudo-BCI algebras are studied and some of their properties are presented.

3

Homomorphisms Between Covering Approximation Spaces

Lang G., Li Q., Guo L.

Fundamenta Informaticae

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2015

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Vol. 138, nr 3

351--371

EN

The introduction of information system homomorphisms has made a substantial contribution to attribute reduction. However, the efforts made on homomorphisms are far from sufficient. This paper further investigates homomorphisms between covering approximation spaces. First, we introduce the concepts of upper and lower homomorphisms as well as homomorphisms in order to study the relationship between covering approximation spaces. Then we present the notions of covering approximation subspaces and product spaces. We also compress covering approximation spaces and covering information systems with the aim of attribute reduction. Afterwards, by utilizing the compressions of the original spaces and systems we compress the dynamic covering approximation spaces and dynamic covering information systems. Several illustrative examples are employed to demonstrate that the homomorphisms provide an effective approach for compressing covering approximation spaces and covering information systems.

4

Compression of Dynamic Fuzzy Relation Information Systems

Cai M., Li Q.

Fundamenta Informaticae

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2015

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Vol. 142, nr 1-4

285--306

EN

The notion of homomorphism, as an important tool for studying the relationship between information systems, has attracted a great deal of attention in recent years, and the authors tend to pay their attention to static information systems in the existing studies. In the present paper, we aim to study homomorphisms between fuzzy relation information systems (FRISs) in dynamic environments, where the terminology of dynamic refers to the fact that the involved information systems need to be updated with time due to the inflow of new information. To be more specific, we firstly examine properties of consistent functions with respect to fuzzy relations and construct homomorphisms between FRISs. Then, we develop incremental mechanisms of computing homomorphisms between dynamic FRISs and illustrate how to construct relation reducts of dynamic FRISs using homomorphisms. Lastly, the experimental results are employed to demonstrate that compressing dynamic FRISs can be simplified significantly with the proposed algorithms.

5

A Unified Definition of Consistent Functions

Zhu P., Xie H., Wen Q.

Fundamenta Informaticae

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2014

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Vol. 135, nr 3

331--340

EN

In recent years, homomorphisms have been exploited to compare the structures and properties of two generalized information systems. Some of these homomorphisms are based on consistent functions, which are a class of special mappings between universal sets. The purpose of this paper is to unify and extend the consistent functions in the literature into the framework of neighborhood systems. After introducing the notion of consistent functions with respect to neighborhood systems, we explore some important properties of the extended consistent functions such as preserving the inverse images and the intersections of neighborhoods. Our results provide a sound basis for further investigating neighborhood systems via homomorphisms.

6

On (…)-centralizers of semiprime rings

Huang S., Haetinger C.

Demonstratio Mathematica

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2012

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Vol. 45, nr 1

29-34

EN

Let R be a semiprime ring with center Z(R) and (…) be a surjective ho-omorphism. In this paper, we prove that T is a (…)-centralizer if one of the following holds: (…).

7

Witruwiańska harmonia form

Kozaczko M.

Czasopismo Techniczne. Architektura

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2009

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R. 106, z. 1-A

345-350

PL

Od niezliczonych pokoleń ludzkie reakcje na bodźce z otoczenia powtarzają się. Przekazywane genetycznie w więźbie aksonów są stale weryfikowane. Miara piękna, odbicie harmonii świata codziennie wdrukowuje się w strukturę mózgu. Przyjemność estetyczna jest zakorzeniona w najstarszych ewolucyjnie częściach mózgu i wyrasta z poczucia zadowolenia, z braku zagrożeń. W naszym umyśle tkwi wzorzec kompozycji idealnej - witruwiańskie venustas.

EN

Experiences, which have been concentrated and layered by thousands of generations in the inner being of man subsequently formed an ideal image of amicable environment, good composition and form. Human mind is a resonator of good form. The work of art built according to rules of such resonance contains visual truth about nature: venustas.

8

A Restarted Strategy for Efficient Subsumption Testing

Kuželka O., Železný F.

Fundamenta Informaticae

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2008

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Vol. 89, nr 1

95-109

EN

We study runtime distributions of subsumption testing. On graph data randomly sampled from two different generative models we observe a gradual growth of the tails of the distributions as a function of the problem instance location in the phase transition space. To avoid the heavy tails, we design a randomized restarted subsumption testing algorithm RESUMER2. The algorithm is complete in that it correctly decides both subsumption and non-subsumption in finite time. A basic restarted strategy is augmented by allowing certain communication between odd and even restarts without losing the exponential runtime distribution decay guarantee resulting from mutual independence of restart pairs. We empirically test RESUMER2 against the state-of-the-art subsumption algorithm Django on generated graph data as well as on the predictive toxicology challenge (PTC) data set. RESUMER2 performs comparably with Django for relatively small examples (tens to hundreds of literals), while for further growing example sizes, RESUMER2 becomes vastly superior.

9

Stability of the equation of ring homomorphisms

Ger R.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

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2007

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Vol. 12

29-35

EN

Let R be a unitary ring and (A,║‧║) stand for a Banach algebra with a unit. In connection with some stability results of R. Badora [1] and D.G. Bourgin [2] concerning the system of two Cauchy functional equations [formula] for mappings f : R→ A, we deal with Hyers-Ulam stability problem for a single equation f(x + y) + f(xy) = f(x) + f(y) + f(x)f(y). The basic question whether or not equation (**) is equivalent to the system (*) has widely been examined by J. Dhombres [3] and the present author in [4] and [5].

10

Retracts of Algebras

Novotny M.

Fundamenta Informaticae

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2007

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Vol. 75, nr 1-4

375-384

EN

Construction of retracts of a general algebra is described by using retracts of mono-unary algebras. Inspirational influence of the theory of Pawlak machines is mentioned.

11

Jordan structure on prime rings with centralizers

Majeed A., Asmaiel H.

Demonstratio Mathematica

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2007

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Vol. 40, nr 3

529-536

EN

Our object in this paper is to study the generalization of Borut Zalar result in [1] on Jordan centralizer of semiprime rings by prove the following result: Let R be a prime of characteristic different from 2, and U be a Jordan ideal of R. If T is an additive mapping from R to itself satisfying the following condition T(ur + ru) = uT(r) + T(r)u, then T(ur) = uT(r), for all r is an element of R, u is an element of U.

12

Weil homomorphism in non-commutative differential spaces

Jagodziński T., Sasin W.

Demonstratio Mathematica

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2005

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Vol. 38, nr 2

507--516

EN

In this paper we construct Weil homomorphism in locally free module over a non-commutative differential space, which is a generalization of Sikorski differential space [6]. We consider real case, but the complex case can be done analogusly.

13

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.

14

Pushouts of partial homomorphisms of partial algebras III: closed-domain closed quomorphisms

Alberich R., Rossello F.

Demonstratio Mathematica

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2004

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Vol. 37, nr 4

773--791

15

Boolean carried homomorphisms in orthomodular lattices

Matousek M.

Demonstratio Mathematica

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2004

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Vol. 37, nr 2

255--265

EN

Let L, L1 be orthomodular lattices. Let us say that a surjective homomorphism f : L - L1 is Boolean carried if for any maximal Boolean subalgebra B1 of L1 there is a maximal Boolean subalgebra B of L such that f(B) = B1. In this note we investigate the class HOMC all L's such that all surjective homomorphisms from L to orthomodular lattices are Boolean carried. We prove as a main result that if L possesses at most countably many infinite maximal Boolean subalgebras then L L HOMC- We also relate the class HOMCto the classes previously studied and provide some model-theoretic propertiesHOMC.

16

On the notion of homomorphism in many-sorted algebras

Pomykała J. A.

Demonstratio Mathematica

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2003

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Vol. 36, nr 3

537--541

17

A note on weakly compact homomorphisms between uniform algebras

Uelger A.

Bulletin of the Polish Academy of Sciences. Mathematics

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1999

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Vol. 47, no 1

1-5

EN

In this paper we present some results about weakly compact homomorphisms from a uniform algebra A into a uniform Algebra B, which is an integral domain. The main result says that if h : A -> B is a weakly compact homomorphism then h*(Sigma[B]) meets one, and only one, Gleason part of the spectrum Sigma[A] of A.

18

Process of contextual nets and their characteristics

Winkowski J.

Prace Instytutu Podstaw Informatyki Polskiej Akademii Nauk

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1998

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nr 847

3-37

PL

Rozważane są uogólnione sieci Petriego zwane sieciami kontekstowymi i procesy takich sieci. Procesy sieci kontekstowych są definiowane jako porządki częściowe z pewna dodatkową strukturą. Definiuje się operacje składania procesów tak, że każdy proces sieci kontekstowej można otrzymać składając procesy odpowiadające miejscom i tranzycjom tej sieci. Definiuje się podobne do macierzy i zwane tablicami charakterystyki procesów i operacje składania tych charakterystyk tak, że odpowiedniość między procesami i charakterystykami jest homomorfizmem.

EN

Generalized Place/Transition Petri nets, called contextual nets, and processes of such nets are considered. Processes of contextual nets are represented as partial orders with some extra structure. Operations of composing processes are defined such that each process of a contextual net can be obtained by combining processes corresponding to places and transitions of this net. Matrix-like characteristics of processes, called tables, and natural operations on such characteristics are defined such that the correspondence between processes and their characteristics is a homomorphism.

19

Processes of contextual nets and their characteristics

Winkowski J.

Fundamenta Informaticae

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1998

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Vol. 36, nr 1

71-101

EN

Generalized Place/Transition Petri nets, called contextual nets, and processes of such nets are considered. Processes of contextual nets are represented as partial orders with some extra structure. Operations of composing processes are defined such that each process of a contextual net can be obtained by combining processes corresponding to places and transitions of this net. Matrix-like characteristics of processes, called tables, and natural operations on such characteristics are defined such that the correspondence between processes and their characteristics is a homomorphism.