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Neutrosophic ${\mathcal N}$-structures and their applications in semigroups
Madad Khan,Saima Anis,Florentin Smarandache,전영배 원광대학교 기초자연과학연구소 2017 ANNALS OF FUZZY MATHEMATICS AND INFORMATICS Vol.14 No.6
The notion of neutrosophic ${\mathcal N}$-structure is introduced, and applied it to semigroup. The notions of neutrosophic ${\mathcal N}$-subsemigroup, neutrosophic ${\mathcal N}$-product and $\varepsilon$-neutrosophic ${\mathcal N}$-subsemigroup are introduced, and several properties are investigated. Conditions for neutrosophic ${\mathcal N}$-structure to be neutrosophic ${\mathcal N}$-subsemigroup are {provided.} Using neutrosophic ${\mathcal N}$-product, characterization of neutrosophic ${\mathcal N}$-subsemigroup is discussed. Relations between neutrosophic ${\mathcal N}$-subsemigroup and $\varepsilon$-neutrosophic ${\mathcal N}$-subsemigroup {are discussed}. We show that the homomorphic preimage of neutrosophic ${\mathcal N}$-subsemigroup is a neutrosophic ${\mathcal N}$-subsemigroup, and the onto homomorphic image of neutrosophic ${\mathcal N}$-subsemigroup is a neutrosophic ${\mathcal N}$-subsemigroup.
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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, $b\;{\in}\;S$.
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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.
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Optimal Campaign Strategies in Fractional-Order Smoking Dynamics
Zeb, Anwar,Zaman, Gul,Jung, Il Hyo,Khan, Madad Verlag der Zeitschrift für Naturforschung 2014 Zeitschrift für Naturforschung. A, A journal Vol.69 No.5
<P>This paper deals with the optimal control problem in the giving up smoking model of fractional order. For the eradication of smoking in a community, we introduce three control variables in the form of education campaign, anti-smoking gum, and anti-nicotive drugs/medicine in the proposed fractional order model. We discuss the necessary conditions for the optimality of a general fractional optimal control problem whose fractional derivative is described in the Caputo sense. In order to do this, we minimize the number of potential and occasional smokers and maximize the number of ex-smokers. We use Pontryagin’s maximum principle to characterize the optimal levels of the three controls. The resulting optimality system is solved numerically by MATLAB.</P>
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