SOOCHOW JOURNAL OF MATHEMATICS

Volume 27, No. 2, pp. 125-138, April 2001




FUZZY IDEALS EXTENSIONS OF SEMIGROUPS$ ^*$


BY


XIANG-YUN XIE



Abstract. We introduce the concepts of the extension of a fuzzy ideal of $ S$ and 3-prime fuzzy ideal of $ S$. We discuss the properties of them and study the relationship between prime ideals of $ S$ and 3-prime fuzzy ideals of $ S$ by means of the extensions of fuzzy ideals of $ S$. Furthermore, let $ S$ be commutative. We prove that a 3-prime semiprime fuzzy ideal of $ S$ can be expressible as the infimum of all prime fuzzy ideals of $ S$ containing it. As a consequence, if $ S$ is a semilattice, then a 3-prime fuzzy ideal of $ S$ is the infimum of prime fuzzy ideals of $ S$ containing it.

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Received October 02, 1998; revised August 08, 1999; revised December 18.

AMS Subject Classification. 04A72, 20M12.

Key words. fuzzy ideal; fuzzy ideal extension, prime fuzzy ideal, semiprime fuzzy ideal, 3-prime fuzzy ideal. $ ^*$This reseach was supported by the NNSF of P. R. China and Guangdong Provincial NSF of P. R. China. The author wishes to thank Prof. Niovi Kehayopulu and Dr. M. Tsingelis whose suggestions improved the paper greatly.