Regularity of Semigroups in Terms of Hybrid Ideals and Hybrid Bi-Ideals

Download PDF


DOI: 10.46793/KgJMat2206.857E


In this paper, we establish some equivalent conditions for a semigroup to be regular and intra-regular, in terms of hybrid ideals and hybrid bi-ideals. We also characterize the left and right simple and the completely regular semigroups utilizing hybrid ideals and hybrid bi-ideals. We show that a semigroup ???? is left simple if and only if it is hybrid left simple. We also prove that if a semigroup ???? is intra-regular, then for each hybrid ideal ˜j μ of ????, we have ˜j μ(r1r2) = ˜j μ(r2r1) for all r1,r2 ∈????.


Semigroup, ideals, regular, hybrid structure, hybrid ideal, hybrid bi-ideal and hybrid product.


[1]   S.  Anis, M.  Khan and Y.  B.  Jun, Hybrid ideals in semigroups, Cogent Mathematics 4(1) (2017), Paper ID 1352117, 12 pages.

[2]   B.  Elavarasan and Y.  B.  Jun, On hybrid ideals and hybrid bi-ideals in semigroups, Iran. J. Math. Sci. Inform. (to appear).

[3]   Y.  B.  Jun, S.  Z.  Sang, and G.  Muhiuddin, Hybrid structures and applications, Annals of Communications in Mathematics 1(1) (2018), 11–25.

[4]   N.  Kehayopulu and M.  Tsingelis, Fuzzy bi-ideals in ordered semigroups, Inform. Sci. 171 (2005), 13–28.

[5]   N.  Kuroki, On fuzzy ideals and fuzzy bi-ideals in semigroups, Fuzzy Sets and Systems 5 (1981), 203–215.

[6]   S.  Lajos, A note on intra-regular semigroups, Proc. Japan Acad. 39(9) (1963), 626–627.

[7]   M.  Mitrović, Regular subsets of semigroups related to their idempotents, Semigroup Forum 70 (2005), 356–360.

[8]   T.  K.  Mukherjee and M.  K.  Sen, Prime fuzzy ideals in rings, Fuzzy Sets and Systems 32 (1989), 337–341.

[9]   D.  Molodtsov, Soft set theory - First results, Comput. Math. Appl. 37 (1999), 19–31.

[10]   A.  Rosenfeld, Fuzzy groups, J. Math. Anal. Appl. 35 (1971), 512–517.

[11]   L.  A.  Zadeh, Fuzzy sets, Information and Control 8 (1965), 338–353.