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The aim of this paper is to introduce two different types of connectedness notions for graded ditopological texture spaces: the connectedness function which gives the grade of connectedness of a set and the connectedness spectrum by means of spectrum idea. Also, the properties of these connectedness notions and their relationships with the connectedness notion in ditopological case are investigated. Further, the relation between these two different types of connectedness notions is studied.

The aim of this paper is to introduce two different types of connectedness notions for graded ditopological texture spaces: the connectedness function which gives the grade of connectedness of a set and the connectedness spectrum by means of spectrum idea. Also, the properties of these connectedness notions and their relationships with the connectedness notion in ditopological case are investigated. Further, the relation between these two different types of connectedness notions is studied. [ABSTRACT FROM AUTHOR]

The centre of attention in this paper deals with j-compactness and j-cocompactness in ditopological texture spaces. Also, j-inadequate and j-coinadequate in ditopological texture spaces are discussed. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the fundamental aspect of H(i) connected ditopological texture space, before proving we develop some basic key of bicontinuity and connectedness in term of ditopological texture space which will helpful in H(i) connected ditopological texture space. We establish some correspondence related to known structure such as bitopological space, fuzzy lattice and topological space.

This paper considers some various categorical aspects of the inverse systems (projective spectrums) and inverse limits described in the category ifPDitop, whose objects are ditopological plain texture spaces and morphisms are bicontinuous point functions satisfying a compatibility condition between those spaces. In this context, the category InvifPDitop consisting of the inverse systems constructed by the objects and morphisms of ifPDitop, besides the inverse systems of mappings, described between inverse systems, is introduced, and the related ideas are studied in a categorical - functorial setting. In conclusion, an identity natural transformation is obtained in the context of inverse systems - limits constructed in ifPDitop and the ditopological infinite products are characterized by the finite products via inverse limits.

The aim of this paper is to introduce and study minimal and maximal open sets, and minimal and maximal closed sets in the context of ditopological texture spaces. Some properties of these notions are presented, and some results on topological spaces are given. [ABSTRACT FROM AUTHOR]

In this paper, we define and study the notions of normal∗ and w-anchored dicovers as important classes of covers, convenient for the texture theory. Especially, a ditopological counterpart is given for the well-known result that in a completely regular topological space, the collection of all normal covers forms a uniformity compatible with the topology. In addition, we describe two kinds of dicovers called divisible and even, in order to characterize the largest di-uniformity on textures, as well as characterizing the full dinormality which is introduced by L.M. Brown and M. Diker (Paracompactness and full normality in ditopological texture spaces, J. Math. Annal. Appl.227 (1998), 144–165). [ABSTRACT FROM AUTHOR]

In this paper, the authors define the notion of ideal on texture spaces. The concept of di-local function is also introduced here by utilizing the families of neighborhood structure for a ditopological texture space. These concepts are discussed with a view to finding new ditopological texture spaces from the original one. Finally, we introduce and give some properties of weakly bicontinuous difunction, a subclass of bicontinuous difunction. [ABSTRACT FROM AUTHOR]

We develop an approach allowing to measure the “quality” of rough approximation of fuzzy sets. It is based on what we call “an approximative quadruple” Q=(L,M,φ,ψ) where L and M are complete lattice commutative monoids and φ:L→M, ψ:M→L are mappings satisfying certain conditions. By realization of this scheme, we get measures of upper and lower rough approximation for L-fuzzy subsets of a set equipped with an M-preoder R:X×X→M. In case R is symmetric, these measures coincide. Basic properties of such measures are studied. Besides, we present an interpretation of measures of rough approximation in terms of LM-fuzzy topologies. [ABSTRACT FROM AUTHOR]

Dynamic relational systems have different forms in literature of rough set theory. They are divided into two parts by Pagliani as synchronic and diachronic dynamics. Synchronic dynamic case is related to the presence of a multi-sources. In the case of diachronic dynamics, it is supposed that changes occur in time. These changes are related to new objects, attributes or attribute values entered into the system. Here, we consider the dynamic systems which are related to the both of these types of dynamics. In a recent paper, the author studied on the results about reduct, definability and quasi-uniformity in dynamic relational systems. Here, the natural generalizations of these results are given. In particular, it is proved that weak and strong definabilities are preserved under pre-images with respect to direlation preserving difunctions between textural dynamic relational systems. Further, the connections between definable sets and the ditopology are determined by ditopologies of reflexive direlations. Then it is given some basic results on ditopologies induced by direlations and direlational quasi-uniformities. Morover, it is discussed on the connections between textural approximation spaces and direlational-quasi uniformities. [ABSTRACT FROM AUTHOR]

This paper considers some various categorical aspects of the inverse systems (projective spectrums) and inverse limits described in the category ifPDitop, whose objects are ditopological plain texture spaces and morphisms are bicontinuous point functions satisfying a compatibility condition between those spaces. In this context, the category InvifPDitop consisting of the inverse systems constructed by the objects and morphisms of ifPDitop, besides the inverse systems of mappings, described between inverse systems, is introduced, and the related ideas are studied in a categorical - functorial setting. In conclusion, an identity natural transformation is obtained in the context of inverse systems - limits constructed in ifPDitop and the ditopological infinite products are characterized by the finite products via inverse limits. [ABSTRACT FROM AUTHOR]

− The main purpose of this paper, is to introduce the notion of ditopological texturespaces with ideal. We study the notions of ?-connected ditopological texture spaces with an idealand ?-connected sets in ditopological texture spaces with ideal. Some new types of connectedness in?-ditopological texture spaces namely, locally ?-connectedness, totally ?-disconnectedness, hyperconnectedness and ?-hyperconnectedness have investigated

In this paper, inverse systems (spectrums) and their inverse limits are described and investigated in the some special subcategories of the category whose objects are ditopological texture spaces and morphisms are the point functions satisfying a compatibility condition, besides bicontinuity. According to that, we especially restrict ourselves to the topological subcategory ifPDitop consisting of point functions as above and ditopological spaces that have plain texturing which is proper, but still quite extensive subclass of texturings. Followed by, so many major properties of inverse systems and limits in the context of category ifPDitop are constructed under the most general conditions. In particular, by considering the full subcategory ifPDicomp of ifPDitop , whose objects are dicompact plain spaces, we acquired various results about the theory of inverse systems consisting of dicompact plain spaces. [ABSTRACT FROM AUTHOR]

The author consider hyperspaces in the setting of textures and ditopological texture spaces. According to that, the definitions of hypertexture, plain hypertexture and hyperspace of ditopological texture space are presented. Then the author obtained some properties of hypertextures in the categorical respect and give some examples of hypertextures. [ABSTRACT FROM AUTHOR]

The basic motivation of the theory of textures is to find a convenient point-set based setting for fuzzy sets. In some recent works on textures show that they also provide a fruitful model for soft fuzzy sets. Further the category of soft fuzzy topological texture spaces is isomorphic to the category of ditopological texture spaces. The first aim of this paper is to introduce and study the notions of semi-compactness and semi-stability in ditopological texture spaces. Secondly, it is given a natural link for these concepts in soft fuzzy topological spaces by using the mentioned isomorphism. [ABSTRACT FROM AUTHOR]

In the present paper, we have continued to study the properties of semi-separation axioms in ditopological texture spaces.We prove that the semi-Ti-spaces (i = 0, 1, 2, 3, 3 1 2 , 4) are productive in ditopological spaces. Finally, semi-separation axioms are given for soft fuzzy topological spaces by using the categorical isomorphism between soft fuzzy topologies and ditopologies. [ABSTRACT FROM AUTHOR]

The author gives a detailed analysis of the relation between the theories of realcompactifications and compactifications in the category of ditopological texture spaces and in the categories of bitopological spaces and topological spaces. [ABSTRACT FROM PUBLISHER]

Topological spaces-such as classifying spaces, configuration spaces and spacetimes-often admit directionality. Qualitative invariants on such directed spaces often are more informative, yet more difficult, to calculate than classical homotopy invariants because directed spaces rarely decompose as homotopy colimits of simpler directed spaces. Directed spaces often arise as geometric realizations of simplicial sets and cubical sets equipped with order-theoretic structure encoding the orientations of simplices and 1-cubes. We prove dual simplicial and cubical approximation theorems appropriate for the directed setting and give criteria for two different homotopy relations on directed maps in the literature to coincide. [ABSTRACT FROM AUTHOR]

he purpose of this paper is to introduce the new concepts namely, α-g-closed, pre-g-closed, semi-g-closed, b-g-closed, β-g-closed, α-g-open, pre-g-open, semi-g-open, b-g-open and β-g-open sets in ditopological texture spaces. The relationships between these classes of sets are also obtained. Also some properties and several characterizations are investigated. [ABSTRACT FROM AUTHOR]

This paper extends the notion of ditopology to the case where openness and closedness are given in terms of a priori unrelated grading functions. The resulting notion of graded ditopology is considered both in the setting of lattices and in that textures, the relation between the two approaches being discussed in detail. Interrelations between graded ditopologies and ditopologies on textures are also studied. [ABSTRACT FROM AUTHOR]

Abstract: The authors extend the notion of real dicompactness from the class of nearly plain textures to the considerably larger class of almost plain textures and show how this may be used to define a notion of realcompactness for Hutton spaces. [Copyright &y& Elsevier]

In this paper the two different concepts namely Ditopological texture spaces and ideals are combined together to dene a new dimension of topology namely ditopological ideal texture spaces. Here we analyse the properties of &agr;-T-open sets and &agr;-T-closed sets in the ditopological ideal texture spaces. [ABSTRACT FROM AUTHOR]

Abstract: Using the characterization of plain textures in terms of posets given by Mustafa Demirci (M. Demirci, Textures and C-spaces, Fuzzy Sets and Systems 158 (11) (2007) 1237–1245), the authors consider the important class of plain ditopological texture spaces and give several new results. [Copyright &y& Elsevier]

Abstract: The authors consider interrelations between the completeness of certain initial di-uniformities and the real dicompactness of completely biregular bi- nearly plain ditopological spaces. Completions and real dicompactifications of almost plain spaces are also considered. [Copyright &y& Elsevier]

In this paper we generalize the notion of α-open and α-closed sets to ditopological texture spaces. We also introduce the concepts of α- bicontinuous difunctions and discuss their properties in detail. [ABSTRACT FROM AUTHOR]

Abstract: Real dicompactifications and dicompactifications of a ditopological texture space are defined and studied. Section 2 considers nearly plain extensions of a ditopological texture space . Spaces that possess a nearly plain extension are shown to have a property, called here almost plainness, that is weaker than that of near plainness, but which shares with near plainness the existence of an associated plain space . Some properties of the class of almost plain ditopological texture spaces are established, a notion of canonical nearly plain extension of an almost plain ditopological texture space, projective and injective pre-orderings and the concept of isomorphism on such canonical nearly plain extensions are defined. In Section 3 the notion of nearly plain extension is specialized to that of real dicompactification and dicompactification, and the spaces that have such extensions are characterized. Working in terms of a specific representation of the canonical real dicompactifications and dicompactifications of a completely biregular bi- almost plain ditopological space, the interrelation between sub-T-lattices of the T-lattice of ω-preserving bicontinuous real mappings on the associated plain space and the real dicompactifications and dicompactifications are investigated. In particular generalizations of the Hewitt realcompactification and Stone–Čech compactification are obtained, and shown to be reflectors for the appropriate categories. [Copyright &y& Elsevier]

The authors characterize the notion of quasi coincident in texture spaces and study the generalization of fuzzy quasi separation axioms defined by [12] to the ditopological texture spaces. [ABSTRACT FROM AUTHOR]

Abstract: A notion of real compactness for completely biregular bi- ditopological texture spaces is defined and studied under the name real dicompactness. In particular it is shown that real dicompact spaces are nearly plain ∗-spaces, and an important characterization is presented. Finally the connection of this work with topological and bitopological real compactness is discussed in a categorical setting. [Copyright &y& Elsevier]

A texturing on a set S is a point separating, complete, completely distributive lattice S of subsets of S with respect to inclusion which contains S, Ø and for which arbitrary meet coincides with intersection and finite joins coincide with union. Then (S, S) is called a texture space. In this paper, a suitable evaluation difunction is defined and an approach for the construction of the Stone-Čech compactification of ditopological texture spaces is given. It is also shown that the Stone-Čech compactification of a topological space can be obtained using highly economic structure of the unit texture. [ABSTRACT FROM AUTHOR]

Abstract: The authors extend the bitopological notion of sequential normality to ditopological texture spaces, and use this notion to state and prove a (pseudo-)dimetrizability theorem. [Copyright &y& Elsevier]

Abstract: A texturing on a set is a point separating, complete, completely distributive lattice of subsets of with respect to inclusion which contains and, for which arbitrary meet coincides with intersection and finite joins coincide with union. The pair is known as a texture space. In this paper, the authors present the concept of embedding for texture spaces and define the notion of difilter on a texture space. Then a Wallman-type compactification is discussed for a class of ditopological texture spaces in terms of so-called difunctions introduced by Brown and his team and it is expressed in the class of molecular weakly bi– Hutton spaces. [Copyright &y& Elsevier]

Abstract: This is the third of three papers which develop various fundamental aspects of the theory of ditopological texture spaces and relate this work with the theory of fuzzy topological spaces. In this paper, the authors define and study basic separation axioms in general ditopological texture spaces, relating these to known separation axioms for bitopological and topological spaces, and to the point-free separation axioms of B. Hutton and I. Reilly for fuzzy topological spaces. [Copyright &y& Elsevier]

Abstract: In the geometric realization of a cubical complex without degeneracies, a □-set, dipaths and dihomotopies may not be combinatorial, i.e., not geometric realizations of combinatorial dipaths and equivalences. When we want to use geometric/topological tools to classify dipaths on the 1-skeleton, combinatorial dipaths, up to dihomotopy, and in particular up to combinatorial dihomotopy, we need that all dipaths are in fact dihomotopic to a combinatorial dipath. And moreover that two combinatorial dipaths which are dihomotopic are then combinatorially dihomotopic. We prove that any dipath from a vertex to a vertex is dihomotopic to a combinatorial dipath, in a non-selfintersecting □-set. And that two combinatorial dipaths which are dihomotopic through a non-combinatorial dihomotopy are in fact combinatorially dihomotopic, in a geometric □-set. Moreover, we prove that in a geometric □-set, the d-homotopy introduced in [M. Grandis, Directed homotopy theory, I, Cah. Topol. Géom. Différ. Catég. 44 (4) (2003) 281–316] coincides with the dihomotopy in [L. Fajstrup, E. Goubault, M. Raussen, Algebraic topology and concurrency, Theoret. Comput. Sci., in press; also technical report, Aalborg University, 1999]. [Copyright &y& Elsevier]

The authors consider difunctions between double-set (intuitionistic) textures, and place the study of double-set textures and ditopological double-set textures in a categorical setting. [ABSTRACT FROM AUTHOR]

In this paper a one-point compactification of a ditopological texture space is defined and a concept of local compactness introduced and shown to be additive in the class of ditopological texture spaces. An application to Hutton spaces is given. Here a compact extension of an Hutton space is obtained and a notion of local compactness is defined and shown to be productive in the class of L-topological spaces, where L is a Hutton algebra. [Copyright &y& Elsevier]

This is the second of three papers which develop various fundamental aspects of the theory of ditopological texture spaces in a categorical setting and present important links with the theory of L-topological spaces. In the first paper in this series, subtitled Basic Concepts, the authors presented the notions of direlation and difunction between textures and introduced the category dfTex, the construct fTex and several related categories. In this paper the category dfDitop of ditopological texture spaces and bicontinuous difunctions is defined and the forgetful functor U : dfDitop → dfTex shown to be topological. Several properties are discussed, including the existence of products and coproducts. An equivalence with the category of classical Hutton spaces is presented and the paper ends with a consideration of L-valued sets and topologies from the viewpoint of ditopological texture spaces. [Copyright &y& Elsevier]