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A study of generalized fuzzy ideals in ordered semigroups

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Abstract

In mathematics, an ordered semigroup is a semigroup together with a partial order that 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 fuzzy generalized sets on ordered semigroups can be developed. Using the notion of “belongingness (∈)” and “quasi-coincidence (q)” of fuzzy points with a fuzzy set, we introduce the concept of an (α, β)-fuzzy left (resp. right) ideal of an ordered semigroup S, where α, β  ∈ {∈, q,  ∈ ∨ q,  ∈ ∧ q} with α ≠  ∈ ∧ q. Since the concept of (∈, ∈∨ q)-fuzzy left (resp. right) ideals is an important and useful generalization of ordinary fuzzy left (resp. right) ideal, we discuss some fundamental aspects of (∈, ∈ ∨ q)-fuzzy left (resp. right) ideals and (\(\overline{\in},\overline{\in}\vee \overline{q}\))-fuzzy left (resp. right) ideals. A fuzzy subset μ of an ordered semigroup S is an (∈, ∈ ∨ q)-fuzzy left (resp. right) ideal if and only if μ t , the level cut of μ is a left (resp. right) ideal of S, for all 0 < t ≤ 0.5 and μ is an (\(\overline{\in},\overline{\in}\vee \overline{q}\))-fuzzy left (resp. right) ideal if and only if μ t is a left (resp. right) ideal of S, for all 0.5 < t ≤ 1. This means that an (∈, ∈ ∨ q)-fuzzy left (resp. right) ideal and (\(\overline{\in},\overline{\in}\vee \overline{q}\))-fuzzy left (resp. right) ideal are generalizations of the existing concept of fuzzy left (resp. right) ideals. Finally, we characterize regular ordered semigroups in terms of (∈, ∈ ∨ q)-fuzzy left (resp. right) ideals.

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Acknowledgments

The authors are very grateful to referees for their valuable comments and suggestions for improving this paper.

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Authors and Affiliations

  1. Department of Mathematics, COMSATS Institute of Information Technology, Khyber Pukhtoon Khwa, Abbottabad, Pakistan

    Asghar Khan

  2. Department of Mathematics Educations and RINS, Gyeonsang National University, Chinju, 660-701, Korea

    Young Bae Jun

  3. Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan

    Muhammad Shabir

Authors
  1. Asghar Khan
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  2. Young Bae Jun
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  3. Muhammad Shabir
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Correspondence to Asghar Khan.

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Khan, A., Jun, Y.B. & Shabir, M. A study of generalized fuzzy ideals in ordered semigroups. Neural Comput & Applic 21 (Suppl 1), 69–78 (2012). https://doi.org/10.1007/s00521-011-0614-6

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  • DOI: https://doi.org/10.1007/s00521-011-0614-6

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