FUZZY INTERIOR $\Gamma$-IDEALS IN ORDERED $\Gamma$-SEMIGROUPS

Khan, Asghar,Mahmood, Tariq,Ali, M. Irfan The Korean Society for Computational and Applied M 2010 Journal of applied mathematics & informatics Vol.28 No.5

In this paper we define fuzzy interior $\Gamma$-ideals in ordered $\Gamma$-semigroups. We prove that in regular(resp. intra-regular) ordered $\Gamma$-semigroups the concepts of fuzzy interior $\Gamma$-ideals and fuzzy $\Gamma$-ideals coincide. We prove that an ordered $\Gamma$-semigroup is fuzzy simple if and only if every fuzzy interior $\Gamma$-ideal is a constant function. We characterize intra-regular ordered $\Gamma$-semigroups in terms of interior (resp. fuzzy interior) $\Gamma$-ideals.

INTUITIONISTIC FUZZY IDEALS IN ORDERED SEMIGROUPS

Asghar Khan,Madad Khan,Saqib Hussain 한국전산응용수학회 2010 Journal of applied mathematics & informatics Vol.28 No.1

We prove that a regular ordered semigroup S is left simple if and only if every intuitionistic fuzzy left ideal of S is a constant function. We also show that an ordered semigroup S is left (resp. right) regular if and only if for every intuitionistic fuzzy left(resp. right) ideal A =<μA,γA>of S we have μA(a) = μA(a²), γA(a) = μA(a²) for every a ∈ S. Further, we characterize some semilattices of ordered semigroups in terms of intuitionistic fuzzy left(resp. right) ideals. In this respect, we prove that an ordered semigroup S is a semilattice of left (resp. right) simple semigroups if and only if for every intuitionistic fuzzy left (resp. right)ideal A = <μA,γA>of S we have μA(a) = μA(a2), γA(a) = μA(a²) and μA(ab) = μA(ba), γA(ab) = γA(ba) for all a.b ∈ S.

INTUITIONISTIC FUZZY IDEALS IN ORDERED SEMIGROUPS

Khan, Asghar,Khan, Madad,Hussain, Saqib The Korean Society for Computational and Applied M 2010 Journal of applied mathematics & informatics Vol.28 No.1

We prove that a regular ordered semigroup S is left simple if and only if every intuitionistic fuzzy left ideal of S is a constant function. We also show that an ordered semigroup S is left (resp. right) regular if and only if for every intuitionistic fuzzy left(resp. right) ideal A = <$\mu_A$, $\gamma_A$> of S we have $\mu_A(a)\;=\;\mu_A(a^2)$, $\gamma_A(a)\;=\;\gamma_A(a^2)$ for every $a\;{\in}\;S$. Further, we characterize some semilattices of ordered semigroups in terms of intuitionistic fuzzy left(resp. right) ideals. In this respect, we prove that an ordered semigroup S is a semilattice of left (resp. right) simple semigroups if and only if for every intuitionistic fuzzy left (resp. right) ideal A = <$\mu_A$, $\gamma_A$> of S we have $\mu_A(a)\;=\;\mu_A(a^2)$, $\gamma_A(a)\;=\;\gamma_A(a^2)$ and $\mu_A(ab)\;=\;\mu_A(ba)$, $\gamma_A(ab)\;=\;\gamma_A(ba)$ for all a, $b\;{\in}\;S$.

Simultaneous reduction and sulfonation of graphene oxide for efficient hole selectivity in polymer solar cells

Asghar Ali,Zuhair S. Khan,Mahmood Jamil,Yaqoob Khan,Nisar Ahmad,S. Ahmed 한국물리학회 2018 Current Applied Physics Vol.18 No.5

We developed sulfonated, reduced graphene oxide (S-RGO) through fuming/concentrated sulfuric acid treatment of graphene oxide (GO) in ambient conditions. It was demonstrated that the optical band gap and electrical conductivity of S-RGO are easily tunable, and depend on the level of reduction and sulfonation of GO. Whereas, reduction and sulfonation were found dependent on SO3 content, acid strength, and gas tightness of the reaction mixture. It's actually the water content of oleum that determines the nature of the final product. The easily adjustable band gap and electrical conductivity suggest that S-RGO can be employed as a potential hole extraction layer (HEL) material for several donor-acceptor systems. For P3HT:PC61BM based inverted polymer solar cells, it was observed that the shape of the JeV curve is tailorable with the choice of HEL. Compared to a 2.75% power conversion efficiency (PCE) attained with PEDOT:PSS, a PCE of 2.80% was achieved with tuned S-RGO. Our results imply that an S-RGO of sufficiently high band gap and conductivity can replace some of the state of the art HEL materials for a host of device applications.

A NEW FORM OF FUZZY GENERALIZED BI-IDEALS IN ORDERED SEMIGROUPS

Khan, Hidayat Ullah,Sarmin, Nor Haniza,Khan, Asghar The Honam Mathematical Society 2014 호남수학학술지 Vol.36 No.3

In several applied disciplines like control engineering, computer sciences, error-correcting codes and fuzzy automata theory, the use of fuzzied algebraic structures especially ordered semi-groups and their fuzzy subsystems play a remarkable role. In this paper, we introduce the notion of (${\in},{\in}{\vee}\bar{q}_k$)-fuzzy subsystems of ordered semigroups namely (${\in},{\in}{\vee}\bar{q}_k$)-fuzzy generalized bi-ideals of ordered semigroups. The important milestone of the present paper is to link ordinary generalized bi-ideals and (${\in},{\in}{\vee}\bar{q}_k$)-fuzzy generalized bi-ideals. Moreover, different classes of ordered semi-groups such as regular and left weakly regular ordered semigroups are characterized by the properties of this new notion. Finally, the upper part of a (${\in},{\in}{\vee}\bar{q}_k$)-fuzzy generalized bi-ideal is defined and some characterizations are discussed.

A study of anti-fuzzy bi-ideals in ordered semigroups

Asghar Khan,Murad ul Islam Khan 원광대학교 기초자연과학연구소 2011 ANNALS OF FUZZY MATHEMATICS AND INFORMATICS Vol.1 No.1

In mathematics, an ordered semigroup is a semigroup together with a partial order which is compatible with the semigroup operation. Ordered semigroups have many applications in the theory of sequential machines, formal languages, computer arithmetics, design of fast adders and error-correcting codes. A theory of anti-fuzzy bi-ideals in ordered semigroups can be developed. Here we define anti-fuzzy bi-ideals of ordered semigroups and give the main theorem which characterizes biideals of ordered semigroups in terms of anti-fuzzy bi-ideals. Then we characterize left and right simple, completely regular and strongly regular ordered semigroups using anti-fuzzy bi-ideals. We also study the characterization of regular and both regular and intra-regular ordered semigroups in terms of anti-fuzzy bi-ideals.

MORE GENERAL FORMS OF (∈; ∈ ∨Q_k) FUZZY FILTERS OF ORDERED SEMIGROUPS

( Asghar Khan ),( Shakoor Muhammad ),( Mohammed M. Khalaf ) 호남수학회 2017 호남수학학술지 Vol.39 No.2

In the paper [Y. B. Jun, B. Davvaz and A. Khan, Filters of ordered semigroups based on the fuzzy points, JIFS 24 (2013) 619-630]. Jun et al. discussed the notion of (∈; ∈ ∨q_k)-fuzzy left (resp., right) filters as a generalization of the notion of (∈; ∈ ∨q)-fuzzy left (resp., right) filters of ordered semigroups. In this article, we try to obtain a more general form that (∈; ∈ ∨q_k)-fuzzy left (resp., right) filters in ordered semigroups. The notion of (∈; ∈ ∨q^δ_k)-fuzzy left (resp., right) filters is discussed, and several properties are investigated. Characterizations of an (∈; ∈ ∨q^δ_k)- fuzzy left (resp., right) filter are established. A condition for an (∈; ∈ ∨q^δ_k)-fuzzy left (resp., right) filter to be a fuzzy left (resp., right) filter is provided. The important achievement of the study with an (∈; ∈ ∨q^δ_k)-fuzzy left (right) filter is that the notion of an (∈; ∈ ∨q_k)-fuzzy left ( right) filter and hence an (∈; ∈ ∨q)-fuzzy left (resp. right) filter are special cases of an (∈; ∈ ∨q^δ_k)-fuzzy left (resp. right) filter, and thus several results in published papers are becoming corollaries of our results obtained in this paper.

MORE GENERAL FORMS OF (∈, ∈ VQ_k) FUZZY FILTERS OF ORDERED SEMIGROUPS

Khan, Asghar,Muhammad, Shakoor,Khalaf, Mohammed M. The Honam Mathematical Society 2017 호남수학학술지 Vol.39 No.2

In the paper [Y. B. Jun, B. Davvaz and A. Khan, Filters of ordered semigroups based on the fuzzy points, JIFS 24 (2013) 619-630]. Jun et al. discussed the notion of (${\in},{\in}{\vee}q_k$)-fuzzy left (resp., right) filters as a generalization of the notion of (${\in},{\in}{\vee}q$)-fuzzy left (resp., right) filters of ordered semigroups. In this article, we try to obtain a more general form that (${\in},{\in}{\vee}q_k$)-fuzzy left (resp., right) filters in ordered semigroups. The notion of (${\in},{\in}{\vee}q_k^{\delta}$)-fuzzy left (resp., right) filters is discussed, and several properties are investigated. Characterizations of an (${\in},{\in}{\vee}q_k^{\delta}$)-fuzzy left (resp., right) filter are established. A condition for an (${\in},{\in}{\vee}q_k^{\delta}$)-fuzzy left (resp., right) filter to be a fuzzy left (resp., right) filter is provided. The important achievement of the study with an (${\in},{\in}{\vee}q_k^{\delta}$)-fuzzy left (right) filter is that the notion of an (${\in},{\in}{\vee}q_k$)-fuzzy left ( right) filter and hence an (${\in},{\in}{\vee}q$)-fuzzy left (resp. right) filter are special cases of an (${\in},{\in}{\vee}q_k^{\delta}$)-fuzzy left (resp. right) filter, and thus several results in published papers are becoming corollaries of our results obtained in this paper.

FUZZY INTERIOR Г-IDEALS IN ORDERED Г-SEMIGROUPS

Asghar Khan,Tariq Mahmood,M. Irfan Ali 한국전산응용수학회 2010 Journal of applied mathematics & informatics Vol.28 No.5

In this paper we de¯ne fuzzy interior Г-ideals in ordered Г-semigroups. We prove that in regular(resp. intra-regular) ordered Г-semigroups the concepts of fuzzy interior Г-ideals and fuzzy Г-ideals coincide. We prove that an ordered Г-semigroup is fuzzy simple if and only if every fuzzy interior Г-ideal is a constant function. We characterize intraregular ordered Г-semigroups in terms of interior (resp. fuzzy interior)Г-ideals.

A New Form of Fuzzy Generalized Bi-Ideals in Ordered Semigroups

( Hidayat Ullah Khan ),( Nor Haniza Sarmin ),( Asghar Khan ) 호남수학회 2014 호남수학학술지 Vol.36 No.3

In several applied disciplines like control engineering, computer sciences, error-correcting codes and fuzzy automata theory, the use of fuzzified algebraic structures especially ordered semigroups and their fuzzy subsystems play a remarkable role. In this paper, we introduce the notion of (-e,-ev-qk,)-fuzzy subsystems of ordered semigroups namely (-e,-ev-qk,)-fuzzy generalized bi-ideals of ordered semigroups. The important milestone of the present paper is to link ordinary generalized bi-ideals and (-e,-ev-qk,)-fuzzy generalized bi-ideals. Moreover, different classes of ordered semigroups such as regular and left weakly regular ordered semigroups are characterized by the properties of this new notion. Finally, the upper part of a (-e,-ev-qk,)-fuzzy generalized bi-ideal is defined and some characterizations are discussed.