Geodesic
Weakly prime and prime fuzzy ideals in ordered semigroups
Lobachevskii journal of mathematics, Tome 27 (2007), pp. 31-40.

Voir la notice de l'article provenant de la source Math-Net.Ru

Our aim is to promote research and the development of fuzzy technology by studying the fuzzy ordered semigroups. The goal is to explain new methodological developments in fuzzy ordered semigroups which will be of growing importance in the future. Intra-regular ordered semigroups play an important role in studying the structure, especially the decomposition of ordered semigroups. In this paper we prove that the fuzzy ideals of an ordered semigroup $S$ are weakly prime if and only if they are idempotent and they form a chain, and that they are prime if and only if $S$ is intra-regular and the fuzzy ideals of $S$ form a chain. Moreover we show that a fuzzy ideal of an ordered semigroup is prime if and only if it is both semiprime and weakly prime and that in commutative ordered semigroups the prime and weakly prime fuzzy ideals coincide. Our results extend the corresponding results on semigroups (without order) given by G. Szász in [15] in case of ordered semigroups using fuzzy sets.
Keywords: intra-regular ordered semigroup, fuzzy subset, fuzzy ideal, weakly prime, prime fuzzy ideal. 
@article{LJM_2007_27_a2,
     author = {N. Kehayopulu},
     title = {Weakly prime and prime fuzzy ideals in ordered semigroups},
     journal = {Lobachevskii journal of mathematics},
     pages = {31--40},
     publisher = {mathdoc},
     volume = {27},
     year = {2007},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/LJM_2007_27_a2/}
}
TY  - JOUR
AU  - N. Kehayopulu
TI  - Weakly prime and prime fuzzy ideals in ordered semigroups
JO  - Lobachevskii journal of mathematics
PY  - 2007
SP  - 31
EP  - 40
VL  - 27
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/LJM_2007_27_a2/
LA  - en
ID  - LJM_2007_27_a2
ER  - 
%0 Journal Article
%A N. Kehayopulu
%T Weakly prime and prime fuzzy ideals in ordered semigroups
%J Lobachevskii journal of mathematics
%D 2007
%P 31-40
%V 27
%I mathdoc
%U http://geodesic.mathdoc.fr/item/LJM_2007_27_a2/
%G en
%F LJM_2007_27_a2
N. Kehayopulu. Weakly prime and prime fuzzy ideals in ordered semigroups. Lobachevskii journal of mathematics, Tome 27 (2007), pp. 31-40. http://geodesic.mathdoc.fr/item/LJM_2007_27_a2/

[1] A. Bargiela, W. Pedrycz, Granular Computing. An Introduction, The Kluwer International Series in Engineering and Computer Science, 717, Kluwer Academic Publishers, Boston, M.A., 2003, ISBN 1-4020-7273-2, xx+452 pp. | MR | Zbl

[2] G. Birkhoff, Lattice Theory, Amer. Math. Soc. Coll. Publ., XXV, Providence, Rhode Island, 1961, xiii+283 pp. ; 1967, vi+418 pp. | MR

[3] A. H. Clifford and G. B. Preston, The Algebraic Theory of Semigroups, Vol. I, Math. Surveys, 7, Amer. Math. Soc., Providence, Rhode Island, 1961, xv+224 pp. | MR | Zbl

[4] L. Fuchs, Partially Ordered Algebraic Systems, Pergamon Press, Oxford-London-New York-Paris; Addison-Wesley Publishing Co., Inc., Reading, Mass.-Palo Alto, Calif.-London, 1963, ix+229 pp. | MR

[5] N. Kehayopulu, “On intra-regular ordered semigroups”, Semigroup Forum, 46:3 (1993), 271–278 | DOI | MR | Zbl

[6] N. Kehayopulu, M. Tsingelis, “Fuzzy sets in ordered groupoids”, Semigroup Forum, 65:1 (2002), 128–132 | DOI | MR | Zbl

[7] N. Kehayopulu, M. Tsingelis, “On fuzzy ordered groupoids-semigroups”, J. Fuzzy Math., 15:2 (2007) (to appear) | MR | Zbl

[8] N. Kehayopulu, M. Tsingelis, “Characterization of some types of ordered semigroups in terms of fuzzy subsets”, Algebra Discrete Math. (to appear)

[9] N. Kehayopulu, M. Tsingelis, “Fuzzy ideals in ordered semigroups”, Lobachevskii J. Math. (to appear)

[10] J. N. Mordeson, D. S. Malik, N. Kuroki, Fuzzy Semigroups, Studies in Fuzziness and Soft Computing, 131, Springer-Verlag, Berlin, 2003, ISBN 3-540-03243-6, x+319 pp. | MR | Zbl

[11] J. N. Mordeson, D. S. Malik, Fuzzy automata and languages, Theory and Applications, Computational Mathematics Series, Chapman and Hall/CRC, Boca Raton, FL, 2002, ISBN 1-58488-225-5, xx+556 pp. | MR | Zbl

[12] W. Pedrycz, F. Gomide, An Introduction to Fuzzy Sets. Analysis and Design, With a foreword by Lotfi A. Zadeh. Complex Adaptive Systems, A Bradford Book, MIT Press, Cambridge, MA, 1998, xxiv+465 pp., ISBN: 0-262-16171-0 | MR | Zbl

[13] M. Petrich, Introduction to semigroups, Merrill Research and Lecture Series, Charles E. Merrill Publishing Co., Columbus, Ohio, 1973, viii+198 pp. | MR | Zbl

[14] G. Szász, “Eine Characteristic der Primidealhalbgrouppen”, Publ. Math. Debrecen, 17 (1970), 209–213 | MR

[15] L. A. Zadeh, “Fuzzy sets”, Inform. Control, 8 (1965), 338–353 | DOI | MR | Zbl