Roughness Applied to Generalized \(\Gamma\)-ideals of Ordered LA \(\Gamma\)-Semigroups

Moin Akhtar Ansari

Abstract


In this paper, we have applied rough set theory to generalized ordered \(\Gamma\)-ideals in ordered LA \(\Gamma\)-semigroups by using a new type of relations called pseudoorder of relations. Based on this relation we have studied properties of rough ideals, rough interior ideals, rough quasi-ideals, rough bi-ideals and rough prime ideals in ordered LA-\(\Gamma\)-semigroups and have shown that lower and upper approximations of an LA-\(\Gamma\)-subsemigroups (resp., left ideals, right ideals, interior ideals, two-sided ideals, quasi-ideals, bi-ideals and prime ideals) in an ordered LA-\(\Gamma\)-semigroup is an LA-\(\Gamma\) subsemigroups (resp., left ideals, right ideals, interior ideals, two-sided ideals, quasi-ideals, bi-ideals and prime ideals).


Keywords


LA-\(\Gamma\)-semigroups; Rough sets; Rough (left, right, quasi, bi, prime and interior)-\(\Gamma\)-ideals

Full Text:

PDF

References


M.I. Ali, T. Mahmood and A. Hussain, A study of generalized roughness in ((in,in_qk))-fuzzy filters of ordered semigroups, Journal of Taibah University For Science 12(2) (2018), pages 1 – 10, DOI: 10.1080/16583655.2018.1451067.

M.A. Ansari and A.N.A. Koam, Rough approximations in KU-algebras, Italian J. of Pure and Applied Mathematics 40 (2018), 679 – 691.

M.A. Ansari and N. Yaqoob, T-rough ideals in ternary semigroups, International Journal of Pure and Applied Mathematics 86(2) (2013), 411 – 424, DOI: 10.12732/ijpam.v86i2.15.

M.A. Ansari, A.N.A. Koam and A. Haider, Rough set theory applied to UP-algebras, Ital. J. Pure Appl. Math. (to appear).

M. Aslam, M. Shabir, N. Yaqoob and A. Shabir, On rough (m,n) bi-ideals and generalized rough (m,n) bi-ideals in semigroups, Ann. Fuzzy Math. Inform. 2(2) (2011), 141 – 150.

R. Biswas and S. Nanda, Rough groups and rough subgroups, Bull. Polish Acad. Sci. Math. 42 (1994), 251 – 254.

T.K. Dutta and N.C. Adhikari, On prime radicals of (Gamma)-semigroups, Bull. Calcutta. Math. Soc. 86(5) (1994), 437 – 444.

Faisal, M. Khan, B. Davvaz and S. Haq, A note on fuzzy ordered AG-groupoids, Journal of Intelligent and Fuzzy Systems 26 (2014), 2251 – 2261, DOI: 10.3233/IFS-130898.

P. Holgate, Groupoids satisfying a simple invertive law, The Math. Stud. 61 (1992), 101 – 106.

M.A. Kazim and M. Naseeruddin, On almost semigroups, Aligarh Bull. Math. 2 (1972), 1 – 7.

N. Kehayopulu, On ordered (Gamma)-semigroups, Scientiae Mathematicae Japonicae Online e-2010, 37 – 43.

M. Khan and Faisal, On fuzzy ordered Abel-Grassmann’s groupoids, J. Math. Res. 3 (2011), 27 – 40, DOI: 10.5539/jmr.v3n2p27.

A. Khan, Faisal, W. Khan and N. Yaqoob, Ordered LA-semigroups in terms of interval valued fuzzy ideals, Journal of Advanced Research in Pure Mathematics 5(1) (2013), 100 – 117.

M. Khan, V. Amjid, G. Zaman and N. Yaqoob, Characterization of ordered (Gamma) abel-grassmann’s Groupoids, Discussiones Mathematicae, General Algebra and Applications 34 (2014), 55 – 73, DOI: 10.7151/dmgaa.1217.

M. Kondo, On the structure of generalized rough sets, Inform. Sci. 176 (2006), 589 – 600, DOI: 10.1016/j.ins.2005.01.001.

N. Kuroki, Rough ideals in semigroups, Inform. Sci. 100 (1997), 139 – 163, DOI: 10.1016/S0020-0255(96)00274-5.

Q. Mushtaq and S.M. Yusuf, On LA-semigroup defined by a commutative inverse semigroups, Math. Bech. 40 (1988), 59 – 62.

Q. Mushtaq and S.M. Yusuf, On LA-semigroups, Aligarh Bull. Math. 8 (1978), 65 – 70.

Z. Pawlak, Rough sets, Int. J. Comput. Inform. Sci. 11 (1982), 341 – 356, DOI: 10.1007/BF01001956.

N. Rehman, C. Park, S.I.A. Shah and A. Ali, On generalized roughness in LA-semigroups, Mathematics 6 (2018), p. 112, DOI: 10.3390/math6070112.

M.K. Sen and N.K. Saha, On (Gamma)-semigroup-I, Bull. Calcutta Math. Soc. 78(3) (1986), 180 – 186.

M.K. Sen, On (Gamma)-semigroups, Algebra and its Applications (New Delhi, 1981), 301-308, in Lecture Notes in Pure and Appl. Math. 91, Dekker, New York (1984).

T. Shah and I. Rehman, On (Gamma)-ideals and bi-(Gamma)-ideals in (Gamma)-AG-groupoid, Int. J. Algebra 4 (2010), 267 – 276.

T. Shah, I. Rehman and A. Ali, On ordering of AG-groupoids, Int. Electronic J. Pure Appl. Math. 2(4) (2010), 219 – 224.

Q.M. Xiao and Z.L. Zhang, Rough prime ideals and rough fuzzy prime ideals in semigroups, Inform. Sci. 176 (2006), 725 – 733, DOI: 10.1016/j.ins.2004.12.010.

N. Yaqoob and R. Chinram, On prime (m,n) bi-ideals and rough prime (m,n) bi-ideals in semigroups, Far East J. Math. Sci. 62(2) (2012), 145 – 159.

N. Yaqoob, M. Aslam and R. Chinram, Rough prime bi-ideals and rough fuzzy prime bi-ideals in semigroups, Ann. Fuzzy Math. Inform. 3(2) (2012), 203 – 211.




DOI: http://dx.doi.org/10.26713%2Fcma.v10i1.1148

Refbacks

  • There are currently no refbacks.


eISSN 0975-8607; pISSN 0976-5905