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Characterizations of Hemirings by the Properties of Their <i>k</i>-Ideals
Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan
Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan
- 1 Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan
- 2 Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan
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Abstract
In this paper we characterize those hemirings for which each k -ideal is idempotent. We also characterize those hemirings for which each fuzzy k -ideal is idempotent. The space of prime k -ideals (fuzzy k -prime k -ideals) is topologized.
KeywordsHemiringFuzzy <i>k</i>-IdealIdempotent k-IdealsPrime IdealsSemiprime Ideals
- H. S. Vandiver, “Note on a Simple Type of Algebra in Which Cancellation Law of Addition Does Not Hold,” Bulletin of the American Mathematical Society, Vol. 40, No. 12, 1934, pp. 914-920. doi:10.1090/S0002-9904-1934-06003-8
- A. W. Aho and J. D. Ullman, “Introduction to Automata Theory, Languages and Computation,” Addison Wesley, Reading, 1976.
- D. B. Benson, “Bialgebras: Some Foundations for Distributed and Concurrent Computation,” Fundamenta Informatica, Vol. 12, 1989, pp. 427-486.
- J. H. Conway, “Regular Algebra and Finite Machines,” Chapman and Hall, London, 1971.
- K. Glazek, “A Guide to Literature on Semirings and Their Applications in Mathematics and Information Sciences with Complete Bibliography,” Kluwer Academic Publishers, Berlin, 2002.
- J. S. Golan, “Semirings and Their Applications,” Kluwer Academic Publishers, Berlin, 1999. doi:10.1007/978-94-015-9333-5
- U. Hebisch and H. J. Weinert, “Semirings: Algebraic Theory and Applications in the Computer Science,” World Scientific, Singapore, 1998. doi:10.1142/3903
- W. Kuich and A. Salomma, “Semirings, Automata, Languages,” Springer Verlag, Berlin, 1986. doi:10.1007/978-3-642-69959-7
- S. Eilenberg, “Automata, Languages and Machines,” Academic Press, New York, 1974.
- E. T. Lee and L. A. Zadeh, “Note on Fuzzy Languages,” Information Sciences, Vol. 1, No. 4, 1969, pp. 421-434. doi:10.1016/0020-0255(69)90025-5
- M. Henriksen, “Ideals in Semirings with Commutative Addition,” Notices of the American Mathematical Society, Vol. 6, 1958, p. 321.
- K. Iizuka, “On the Jacobson Radial of a Semiring,” Tohoku Mathematical Journal, Vol. 11, 1959, pp. 409-421.
- D. R. LaTorre, “On h-Ideals and k-Ideals in Hemirings,” Publicationes Mathematicae (Debrecen), Vol. 12, 1965, pp. 219-226.
- L. A. Zadeh, “Fuzzy Sets,” Infection Control, Vol. 8, No. 3, 1965, pp. 338-353. doi:10.1016/S0019-9958(65)90241-X
- J. Ahsan, K. Saifullah and M. Farid Khan, “Fuzzy Semirings,” Fuzzy Sets Systems, Vol. 60, No. 3, 1993, pp. 309-320. doi:10.1016/0165-0114(93)90441-J
- J. Ahsan, “Semirings Characterized by Their Fuzzy Ideals,” Journal of Fuzzy Mathematics, Vol. 6, 1998, pp. 181-192.
- M. Akram and W. A. Dudek, “Intuitionistic Fuzzy Left k-Ideals of Semirings,” Soft Computing, Vol. 12, No. 9, 2008, pp. 881-890. doi:10.1007/s00500-007-0256-x