Characterizations of ordered Γ-Abel-Grassmann's groupoids

Madad Khan; Venus Amjid; Gul Zaman; Naveed Yaqoob

Discussiones Mathematicae - General Algebra and Applications (2014)

  • Volume: 34, Issue: 1, page 55-73
  • ISSN: 1509-9415

Abstract

top
In this paper, we introduced the concept of ordered Γ-AG-groupoids, Γ-ideals and some classes in ordered Γ-AG-groupoids. We have shown that every Γ-ideal in an ordered Γ-AG**-groupoid S is Γ-prime if and only if it is Γ-idempotent and the set of Γ-ideals of S is Γ-totally ordered under inclusion. We have proved that the set of Γ-ideals of S form a semilattice, also we have investigated some classes of ordered Γ-AG**-groupoid and it has shown that weakly regular, intra-regular, right regular, left regular, left quasi regular, completely regular and (2,2)-regular ordered Γ-AG**-groupoids coincide. Further we have proved that every intra-regular ordered Γ-AG**-groupoid is regular but the converse is not true in general. Furthermore we have shown that non-associative regular, weakly regular, intra-regular, right regular, left regular, left quasi regular, completely regular, (2,2)-regular and strongly regular Γ-AG*-groupoids do not exist.

How to cite

top

Madad Khan, et al. "Characterizations of ordered Γ-Abel-Grassmann's groupoids." Discussiones Mathematicae - General Algebra and Applications 34.1 (2014): 55-73. <http://eudml.org/doc/270662>.

@article{MadadKhan2014,
abstract = {In this paper, we introduced the concept of ordered Γ-AG-groupoids, Γ-ideals and some classes in ordered Γ-AG-groupoids. We have shown that every Γ-ideal in an ordered Γ-AG**-groupoid S is Γ-prime if and only if it is Γ-idempotent and the set of Γ-ideals of S is Γ-totally ordered under inclusion. We have proved that the set of Γ-ideals of S form a semilattice, also we have investigated some classes of ordered Γ-AG**-groupoid and it has shown that weakly regular, intra-regular, right regular, left regular, left quasi regular, completely regular and (2,2)-regular ordered Γ-AG**-groupoids coincide. Further we have proved that every intra-regular ordered Γ-AG**-groupoid is regular but the converse is not true in general. Furthermore we have shown that non-associative regular, weakly regular, intra-regular, right regular, left regular, left quasi regular, completely regular, (2,2)-regular and strongly regular Γ-AG*-groupoids do not exist.},
author = {Madad Khan, Venus Amjid, Gul Zaman, Naveed Yaqoob},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {ordered Γ-AG-groupoids; Γ-ideals; regular Γ-AG**-groupoids},
language = {eng},
number = {1},
pages = {55-73},
title = {Characterizations of ordered Γ-Abel-Grassmann's groupoids},
url = {http://eudml.org/doc/270662},
volume = {34},
year = {2014},
}

TY - JOUR
AU - Madad Khan
AU - Venus Amjid
AU - Gul Zaman
AU - Naveed Yaqoob
TI - Characterizations of ordered Γ-Abel-Grassmann's groupoids
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2014
VL - 34
IS - 1
SP - 55
EP - 73
AB - In this paper, we introduced the concept of ordered Γ-AG-groupoids, Γ-ideals and some classes in ordered Γ-AG-groupoids. We have shown that every Γ-ideal in an ordered Γ-AG**-groupoid S is Γ-prime if and only if it is Γ-idempotent and the set of Γ-ideals of S is Γ-totally ordered under inclusion. We have proved that the set of Γ-ideals of S form a semilattice, also we have investigated some classes of ordered Γ-AG**-groupoid and it has shown that weakly regular, intra-regular, right regular, left regular, left quasi regular, completely regular and (2,2)-regular ordered Γ-AG**-groupoids coincide. Further we have proved that every intra-regular ordered Γ-AG**-groupoid is regular but the converse is not true in general. Furthermore we have shown that non-associative regular, weakly regular, intra-regular, right regular, left regular, left quasi regular, completely regular, (2,2)-regular and strongly regular Γ-AG*-groupoids do not exist.
LA - eng
KW - ordered Γ-AG-groupoids; Γ-ideals; regular Γ-AG**-groupoids
UR - http://eudml.org/doc/270662
ER -

References

top
  1. [1] R. Chinram and K. Tinpun, A note on minimal bi-ideals in ordered Γ-semigroups, International Math. Forum 4 (1) (2009) 1-5. Zbl1179.06006
  2. [2] K. Hila, Filters in ordered Γ-semigroups, Rocky Mountain J. Math. 41 (1) (2011) 189-203. doi: 10.1216/RMJ-2011-41-1-189 Zbl1211.06015
  3. [3] K. Hila, On quasi-prime, weakly quasi-prime left ideals in ordered Γ-semigroups, Math. Slovaca 60 (2) (2010) 195-212. doi: 10.2478/s12175-010-0006-x Zbl1240.06073
  4. [4] K. Hila and E. Pisha, On bi-ideals on ordered Γ-semigroups I, Hacettepe J. Math. and Stat. 40 (6) (2011) 793-804. Zbl1266.06031
  5. [5] K. Hila and E. Pisha, On lattice-ordered rees matrix Γ-semigroups, Annals of the Alexandru Ioan Cuza University - Mathematics LIX (1) (2013) 209-218. doi: 10.2478/v10157-012-0033-8 Zbl1289.06038
  6. [6] K. Hila and E. Pisha, Characterizations on ordered Γ-semigroups, Inter. J. Pure and Appl. Math. 28 (3) (2006) 423-440. Zbl1115.06007
  7. [7] P. Holgate, Groupoids satisfying a simple invertive law, The Math. Student 61 (1992) 101-106. Zbl0900.20160
  8. [8] A. Iampan, Characterizing ordered bi-Ideals in ordered Γ-semigroups, Iranian J. Math. Sci. and Inf. 4 (1) (2009) 17-25. Zbl1301.06052
  9. [9] A. Iampan, Characterizing ordered quasi-ideals of ordered Γ-semigroups, Kragujevac J. Math. 35 (1) (2011) 13-23. 
  10. [10] M.A. Kazim and M. Naseeruddin, On almost semigroups, The Aligarh Bull. Math. 2 (1972) 1-7. 
  11. [11] N. Kehayopulu and M. Tsingelis, Regular ordered semigroups in terms of fuzzy subsets, Inf. Sci. 176 (2006) 3675-3693. doi: 10.1016/j.ins.2006.02.004 Zbl1141.06013
  12. [12] M. Khan, Some studies in AG*-groupoids, Ph.D., thesis (Quaid-i-Azam University, Islamabad, Pakistan, 2008). 
  13. [13] M. Khan and N. Ahmad, Characterizations of left almost semigroups by their ideals, J. Adv. Res. Pure Math. 2 (2010) 61-73. doi: 10.5373/jarpm.357.020210 
  14. [14] M. Khan, T. Asif and Faisal, Intra-regular left almost semigroups characterized by their anti fuzzy ideals, J. Math. Res. 4 (2010) 100-110. Zbl1207.20066
  15. [15] Y.I. Kwon and S.K. Lee, On weakly prime ideals of ordered Γ-semigroups, Comm. Korean Math. Society 13 (2) (1998) 251-256. Zbl0968.06501
  16. [16] Y.I. Kwon, Characterizations of regular ordered Γ-semigroups II, Far East J. Math. Sci. 11 (3) (2003) 281-287. Zbl1041.06007
  17. [17] Y.I. Kwon and S.K. Lee, Some special elements in ordered Γ-semigroups, Kyungpook Math. J. 35 (1996) 679-685. Zbl0986.06500
  18. [18] Y.I. Kwon and S.K. Lee, The weakly semi-prime ideals of po-Γ-semigroups, Kangweon Kyungki Math. J. 5 (2) (1997) 135-139. 
  19. [19] Q. Mushtaq and M. Khan, Ideals in left almost semigroups, Proceedings of 4th International Pure Mathematics Conference (2003) 65-77. 
  20. [20] Q. Mushtaq and M.S. Kamran, On LA-semigroups with weak associative law, Scientific Khyber 1 (1989) 69-71. 
  21. [21] Q. Mushtaq and S.M. Yousuf, On LA-semigroups, Aligarh Bull. Math. 8 (1978) 65-70. 
  22. [22] Q. Mushtaq and S.M. Yousuf, On LA-semigroup defined by a commutative inverse semigroup, Math. Bech. 40 (1988) 59-62. Zbl0692.20053
  23. [23] M. Naseeruddin, Some studies in almost semigroups and flocks, Ph.D., thesis (Aligarh Muslim University, Aligarh, India, 1970). 
  24. [24] P.V. Protić and N. Stevanović, AG-test and some general properties of Abel-Grassmann's groupoids, Pure Math. and Appl. 4 (6) (1995) 371-383. Zbl0858.20058
  25. [25] T. Shah and I. Rehman, On M-systems in Γ-AG-groupoids, Proceedings of the Pakistan Academy of Sciences 47 (1) (2010) 33-39. 
  26. [26] N. Stevanović and P.V. Protić, Composition of Abel-Grassmann's 3-bands, Novi Sad J. Math. 2 (34) (2004) 175-182. Zbl1060.20056

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.