Generalized roughness in fuzzy filters and fuzzy ideals with thresholds in ordered semigroups
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Generalized roughness in fuzzy filters and fuzzy ideals with thresholds in ordered semigroups Tahir Mahmood1 Azmat Hussain1
· Muhammad Irfan Ali2 ·
Received: 13 October 2017 / Revised: 20 February 2018 / Accepted: 20 March 2018 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2018
Abstract In the present paper, concept of roughness for fuzzy filters with thresholds (u 1 , u 2 ) in ordered semigroups is introduced. Then, this concept is extended to fuzzy bi-filters with thresholds and fuzzy quasi-filters with thresholds. Further approximations of fuzzy ideals with thresholds, fuzzy bi-ideals with thresholds, and fuzzy interior ideals with thresholds are studied. Moreover, this concept is applied to study approximations of fuzzy quasi-ideals with thresholds and semiprime fuzzy quasi-ideals with thresholds. Keywords Rough sets · Fuzzy set · Ordered Semigroups · Approximation of fuzzy filters with thresholds · Approximation of fuzzy ideals with thresholds Mathematics Subject Classification 08A72 · 34C41 · 06F05
1 Introduction To deal with uncertainty, rough set theory and fuzzy set theory are two very popular notions. Although these theories are very distinct in nature, yet it can be combined in a very productive manner.
Communicated by Marcos Eduardo Valle.
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Tahir Mahmood [email protected] Muhammad Irfan Ali [email protected] Azmat Hussain [email protected]
1
Department of Mathematics and Statistics, Faculty of Basic and Applied Sciences, International Islamic University, Islamabad, Pakistan
2
Department of Mathematics, Islamabad Model College for Girls, F-6/2, Islamabad, Pakistan
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T. Mahmood et al.
The rudimentary concepts of fuzzy sets are given by Zadeh (1965) and furnished a model for generalizing copious of the concepts and numerous vestigial algebraic notions. Initially, Rosenfeld gave the concept of fuzzy subgroups (Rosenfeld 1971). Kuroki (1991) initiated the concept of fuzzy semigroup. The idea of fuzzy ordered groupoids and fuzzy ordered semigroups was innovated by Kehayopulu and Tsingelis (1999, 2002). They pioneered the idea of fuzzy filters in ordered semigroups and also acquainted some vestigial properties of fuzzy filters and prime fuzzy ideals which have been discussed in Kehayopulu and Tsingelis (2002). A type of generalized fuzzy filters of R0 algebra is given by Ma et al. (2009) and furnished some properties of this notion. Kuroki studied fuzzy ideals, fuzzy bi-ideals, fuzzy quasi-ideals, and semiprime ideals in semigroups, please see (Kuroki 1979, 1981, 1993). In Bhakat and Das (1996), Bhakat and Das initiated the concept of (∈, ∈ ∨q)-fuzzy subgroups. They showed that fuzzy subgroups introduced by Rosenfeld are a particular case of (∈, ∈ ∨q)fuzzy subgroups. (∈, ∈ ∨q) fuzzy bi-ideals are studied by Kazanci and Yamak (2008) . A further generalization of (∈, ∈ ∨q)-fuzzy subgroups is given in Yuan et al. (2003), where concept of fuzzy subgroups with thresholds is introduced. In Kazanci and Yamak (2008), idea of fuzzy bi-ideals with thresholds in s
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