Relative bi-ideals and relative quasi ideals in ordered semigroups

In this paper, after introducing the notion of relative bi-ideals and relative quasi ideals in ordered semigroups, some important properties of these bi-ideals and quasi ideals are studied. Then relatively prime and relatively weakly semiprime bi-ideals are defined and some vital results have been proved. We also define relative regularity and relative intra-regularity of an ordered semigroup and prove some results based on the connection among intra-regularity of an ordered semigroup, relative quasi and relative bi-ideals of that ordered semigroup. Finally some important results connecting relative regularity, relatively prime bi-ideals and relatively weakly semiprime bi-ideals of an ordered semigroup have also been obtained.

Keywords:

Ordered semigroups, ideals, relative ideals, bi-ideals, relative bi-ideals, relative quasi ideals relatively prime and relatively weakly semiprime bi-ideals, relative regularity and relative intra-regularity,

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[1] M.F. Ali, N.M. Khan and A. Mahboob, Relative ideals in ordered semigroups, sub- mitted for publication.
[2] N.G. Alimov, On ordered semigroups, Izvestiya Akad. Nauk SSSR. 14, 569–576, 1950.
[3] A.H. Clifford, Totally ordered commutative semigroups, Bull. Amer. Math. Soc. 64, 305–316, 1958.
[4] A.H. Clifford and G.B. Preston, The algebraic theory of semigroups. Vol. I., Mathe- matical Surveys, No. 7, American Mathematical Society, Providence, R.I., 1961.
[5] L. Fuchs, Partially ordered algebraic systems, Pergamon Press, 113–115, 1963.
[6] R.A. Good and D.R. Hughes, Associative groups for a semigroup, Bull. Amer. Math. Soc. 58, 79–81, 1958.
[7] Y.V. Hion, Ordered semigroups, Izvestiya Akad. Nauk SSSR. 21, 209–222, 1957.
[8] N. Kehayopulu, On weakly prime ideals of ordered semigroups, Math. Japonica, 35( 6), 1051–1056, 1990.
[9] N. Kehayopulu, On prime, weakly prime ideals in ordered semigrous, Semigroup Fo- rum, 44, 341–346, 1992.
[10] N. Kehayopulu, On Regular duo ordered semigroups, Math. Japonica, 37, 535–540, 1992.
[11] N. Kehayopulu, S. Lajos and M. Singelis, On intra-regular ordered semigroups, PU. M. A. 4, 317–327, 1993.
[12] S. Lajos and G. Szasz, On characterizations of certain classes of semigroups, Publ. Math. Debecen, 25, 225–227, 1978.
[13] S. Lajos, Bi-ideals in semigroups I, A survey PU. M. A. Ser. A, 2, 3–4, 1991.
[14] D.M. Lee and S.K. Lee, On Intra-regular Ordered Semigroups, Kangweon-Kyungki Math. Jour. 14 (1), 95–100, 2006.
[15] H.J. le Roux, A note on Prime and Semiprime Bi-ideals of Rings, Kyungpook Math. J. 35, 243–247, 1995.
[16] Renáta Hrmová, Relative ideals in semigroups, Matematický Casopis, 17 (3), 206– 223, 1967.
[17] R. Saritha, Prime and Semiprime Bi-ideals in Ordered Semigroups, Int. J. Algebra, 7 (17), 839–845, 2013.
[18] O. Steinfeld, On quotients and prime ideals, Acta. Math. Acad. Sci. Hung. 4, 289–298, 1953.
[19] O. Steinfeld, Quasi ideals in Rings and Semigroups, Akademiaikiado, Budapest, 1978.
[20] G. Szász, Eine Charakteristik der Primidealhalbgruppen, Publ. Math. Debrecen, 17, 209–213, 1970.
[21] A.P.J. van der Walt, Prime and Semiprime Bi-ideals, Quaest. Math. 5, 341–345, 1983.
[22] A.D. Wallace, Relative ideals in semigroups I, Colloq. Math. 9, 55–61, 1962.

Hacettepe Journal of Mathematics and Statistics-Cover

Yayın Aralığı: Yılda 6 Sayı
Başlangıç: 2002
Yayıncı: Hacettepe Üniversitesi Fen Fakultesi

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