Interior and closure operators on texture spaces—II: Dikranjan–Giuli closure operators and Hutton algebras

Diker, Murat; Dost, Şenol; Uğur, Ayşegül Altay

doi:10.1016/j.fss.2009.10.009

Cited by 6 publications

(11 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Recent works on textures show that they also provide a useful model for rough set theory [8]. In this paper, we show that i-c spaces (interior-closure texture spaces) studied in [12, 13] can be regarded as a textural rough set systems on a single universe. Then we consider the approaches containing direlations and dicovers for textural rough sets and we give some basic results related to direlations and dicovers.…”

mentioning

confidence: 90%

“…Let (U, U) be a texture. In [12], any two set valued mappings int, cl : U → U are said to be generalized interior and closure operators respectively, and the quadruple (U, U, int, cl) is called to be a generalized interior-closure space (gic-space). Also, the textural lower-upper approximation operators which are defined by Diker [8], are set valued mappings.…”

Section: Is Called Bicontinuous If It Is Both Continuous and Cocontinmentioning

confidence: 99%

“…Note that the operators int and cl which satisfy (L 1 )-(H 1 ) are called grounded [12]. Because of the operators int and cl are sublinear, they are isotone by [ …”

Section: Is Called Bicontinuous If It Is Both Continuous and Cocontinmentioning

confidence: 99%

“…expansive) and sublinear, then the universe with these operators is called an i-c space. Recall that i-c spaces (interior-closure texture spaces) and bicontinuous difunctions form a category denoted by dfIC [12]. Likewise, complemented ic spaces and bicontinuous complemented difunctions form a category denoted by cdfIC.…”

mentioning

confidence: 99%

See 3 more Smart Citations

A categorical view of textural approximation spaces

Dost

2015

Quaestiones Mathematicae

View full text Add to dashboard Cite

The basic motivation of the theory of textures is to find a convenient point-set based setting for fuzzy sets. Recent works on textures show that they also provide a useful model for rough set theory. In this paper, we show that i-c spaces (interior-closure texture spaces) can be regarded as a textural rough set systems on a single universe. Then we consider the approaches containing direlations and dicovers for textural rough sets and we give some basic results related to direlations and dicovers. Considering the discrete textures we discuss on these results for rough sets based on relations and coverings. Finally, we prove that the category of topological spaces and continuous functions is isomorphic to the category of (covering) approximation spaces and continuous functions. (2010): 54A40, 06A15, 18B30. Mathematics Subject Classification Introduction.Rough set theory is an extension of set theory as a mathematical approach to information systems represented by equivalence relations [15]. Generalizations of rough sets using arbitrary relations or coverings provide important real-life applications in natural computing [18][19][20][21].The starting point of the theory of textures is a point-set based setting for fuzzy sets [2,4]. Recent works on textures show that they also provide a useful model for rough set theory [8]. In this paper, we show that i-c spaces (interior-closure texture spaces) studied in [12, 13] can be regarded as a textural rough set systems on a single universe. Then we consider the approaches containing direlations and dicovers for textural rough sets and we give some basic results related to direlations and dicovers. Finally, considering the discrete textures we discuss on these results for rough sets based on relations and coverings.If interior and closure operators on a universe are grounded, isotonic, contractive (resp. expansive) and sublinear, then the universe with these operators is called an i-c space. Recall that i-c spaces (interior-closure texture spaces) and bicontinuous difunctions form a category denoted by dfIC [12]. Likewise, complemented ic spaces and bicontinuous complemented difunctions form a category denoted by cdfIC. Here, we work on i-c spaces which we call i-c * spaces whose interior and closure operators are grounded and sublinear where sublinearity holds for

show abstract

mentioning

confidence: 90%

Section: Is Called Bicontinuous If It Is Both Continuous and Cocontinmentioning

confidence: 99%

“…Note that the operators int and cl which satisfy (L 1 )-(H 1 ) are called grounded [12]. Because of the operators int and cl are sublinear, they are isotone by [ …”

Section: Is Called Bicontinuous If It Is Both Continuous and Cocontinmentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

A categorical view of textural approximation spaces

Dost

2015

Quaestiones Mathematicae

View full text Add to dashboard Cite

show abstract

“…This notion, originally designed to help characterize epimorphisms in subcategories of topological spaces and to determine whether such subcategories are cowell-powered (so that every object X allows for only a set of nonisomorphic epimorphisms with domain X), has enjoyed considerable attention; see in particular the monographs [23,8]. Its applications range from topology to algebra and theoretical computer science; see, for example, [22,24,6,14,19,20]. What is the categorically dual notion of closure operator?…”

Section: Introductionmentioning

confidence: 99%

Dual closure operators and their applications

Dikranjan

Tholen

2015

Journal of Algebra

View full text Add to dashboard Cite

Departing from a suitable categorical description of closure operators, this paper dualizes this notion and introduces some basic properties of dual closure operators. Usually these operators act on quotients rather than subobjects, and much attention is being paid here to their key examples in algebra and topology, which include the formation of monotone quotients (Eilenberg-Whyburn) and concordant quotients (Coffins). In fair categorical generality, these constructions are shown to be factors of the fundamental correspondence that relates connectecinesses and disconnectednesses in topology, as well as torsion classes and torsion-free classes in algebra. Depending on a given cogenerator, the paper also establishes a non-trivial correspondence between closure operators and dual closure operators in the category of R-modules. Dual closure operators must be carefully distinguished from interior operators that have been studied by other author

show abstract

Cowellpoweredness of Some Categories of Quasi-Uniform Spaces

Dikranjan

Künzi

2018

Appl Categor Struct

View full text Add to dashboard Cite

Interior and closure operators on texture spaces—II: Dikranjan–Giuli closure operators and Hutton algebras

Cited by 6 publications

References 15 publications

A categorical view of textural approximation spaces

A categorical view of textural approximation spaces

Dual closure operators and their applications

Cowellpoweredness of Some Categories of Quasi-Uniform Spaces

Contact Info

Product

Resources

About