A Comparative Study of Anti-rectangular AG-groupoids and Commutative Semigroups


  • I. Ahmad University of Malakand
  • A. Rauf University of Malakand
  • M. Khan University of Malakand


In this paper, anti-rectangular AG-groupoids are studied. Some properties of anti-rectangular AGgroupoid
are investigated via a concise study of its super classes. Furthermore, various conditions are
explored under which an anti-rectangular AG-groupoid becomes commutative and hence a commutative
semigroup. Alternatively, the paper is a pleasant contribution to the theory of commutative semigroups.
Moreover, using the latest computational techniques of Prove-9 and Mace-4 a variety of examples and
counterexamples are provided to depict various produced results.

Author Biographies

I. Ahmad, University of Malakand

Department of Mathematics,  Chakdara Dir (L)

A. Rauf, University of Malakand

Department of Mathematics,  Chakdara Dir (L)

M. Khan, University of Malakand

Department of Mathematics,  Chakdara Dir (L)



M.A. Kazim and M. Naseeruddin, “On almost semigroupâ€, Portugaliae Mathematics, vol. 2, pp. 1-7, 1972.

M. Naseeruddin, “Some studies on almost semigroups and flocksâ€, PhD Thesis, The Aligarh Muslim University, Aligarh, India, 1970.

J.R. Cho, Pusan, J. Jezek and T. Kepka, “Paramedial groupoidsâ€, Czech. Math. J., vol. 49, No. 124, pp. 277-290, 1996.

P. Holgate, “Groupoid satisfying a simple invertive lawâ€, Mathematics Students, vol. 61, pp. 101-106, 1992.

M. Shah, “A theoretical and computational investigation of AG groupsâ€, PhD Thesis, Qauid-i-Azam University Islamabad, Pakistan, Available at http://prr.hec.gov.pk/jspui/handle/123456789/14142, (2012).

I. Ahmad, Amanullah and M. Shah, “Fuzzy AG-subgroupsâ€, Life Sci. J., vol. 9, no. 4, pp. 3931-3936, 2012.

I. Ahmad, I. Ahmad and M. Rashad, “A study of anti-commutativity in AG-Groupoidsâ€, Punjab University Journal of Mathematics, vol. 48, no. 1, pp. 99-109, 2016.

Amanullah, M. Rashad and I. Ahmad, “Abel-Grassmann’s groupoids of modulo matricesâ€, Mehran University Research Journal of Engineering & Technology, vol. 35, no. 1, pp. 63-70, 2016.

Amanullah, M. Rashad, I. Ahmad and M. Shah, “On modulo AG-groupoidsâ€, Journal of Advances in Mathematics (JAAM), vol. 8, no. 3, pp. 1606-1613, 2014. [10] F. Karaslan, I. Ahmad and Amanullah, “Bipolar soft groupsâ€, J. Intell. Fuzzy Syst., vol. 31, no. 651-662, 2016.

A. Distler, M. Shah and V. Sorge, “Enumeration of AG-groupoidâ€, Intelligent Computer Mathematics Joint Proceeding of Calculemus, vol. 6824 of Lecture, 2011.

A. Distler, M. Shah and V. Sorge, “Enumeration of AG-groupoidsâ€, Lect. Notes Comput. Sci, vol. 6824/2011, pp. 1-14, 2011.

M. Shah, I. Ahmad and A. Ali, “Discovery of new classes of AG-gruopoidâ€, Research Journal of Recent Science, vol. 1, no. 11, pp. 47-49, 2012.

M. Shah, A. Ali and I. Ahmad, “On introduction of new classes of AG-groupoidsâ€, Res. J. Recent Sci., vol. 2, no. 1, pp. 67-70, 2013.

M. Rashad, “Classification and investigation of some classes of AG-groupoidsâ€, PhD Thesis, University of Malakand, Chakdara, Pakistan, 2015.

I. Ahmad, M. Rashad and M. Shah, “Some Properties of AG*-groupoidâ€, Res. J. Recent Sci., vol. 2, no. 4, pp. 91-93, 2013.

I. Ahmad, M. Rashad and M. Shah, “Some new result on T1, T2 and T4-AG-groupoidsâ€, Res. J. of Recent Sci., vol. 2, no. 3, pp. 64-66, 2013.

M. Rashad, I. Ahmad, Amanullah and M. Shah, “On relations between right alternative and nuclear square AG-groupoidsâ€, Int. Math. Forum, vol. 8, no. 5, pp. 237-243, 2013.

M. Rashad, I. Ahmad, M. Shah and A.B. Saeid, “Enumeration of Bi-commutative AG-groupoidsâ€, Journal of Siberian Federal University Mathematics & Physics, vol. 13, no. 3, pp. 314-330, DOI: 10.17516/1997-1397-2020-13-3-314-330, (2020).

M. Rashad, I. Ahmad, “A Note on Unar LA-Semigroupâ€, Punjab University Journal of Mathematics (ISSN 1016-2526), vol. 50, no. 4, 2018.

M. Shah and I. Ahmad, “Paramedial and Bol* Abel Grassmann groupoidsâ€, The Nucleus, vol. 52, no. 3, pp. 130-137, 2015.

M. Shah and I. Ahmad, “Paramedial and Bol* Abel-Grassmann’s groupoidsâ€, The Nucleus 52, no. 3, no. 130-137, 2015.

M. Iqbal and I. Ahamd, “Some further investigation on cyclic associative AG-groupoidâ€, Proc. of the Pakistan Academy of Sciences, (A. Physical and Computational Sciences), vol. 53, no. 3, pp. 337–349, 2016.

M. Rashad, I. Ahmad, M. Shah and Amanullah, “Some properties of Stein AG-groupoids and Stein-testâ€, Sindh University Research Journal (Sci. Series), vol. 48, no. 3, pp. 679-684, 2016.

I. Ahmad, Rohi Naz, M. Rashad and A. Khan, “A comparative study of AG*-groupoidsâ€, The Nucleus, 53, no. 4, pp. 269-274, 2016.

M. Iqbal and I. Ahmad, “Ideals in CA-AG-groupoidsâ€, Indian J. Pure Ap. Mat., Ms. No. IJPA-D-16-01038, vol. 49, no. 2, 2018.

M. Khan, F. Smarandache and S. Anis, “Theory of Abel-Grassmann groupoidsâ€, The Educational Publishers, INC., Ohio, USA, 2015.

Q. Mushtaq and M. Khan, “Topological Structure on Abel Grassmann's groupoidsâ€, ArXive: 0904.1650v1 [math.GR] 10 Apr (2009).

M. Khan, “Study of anti-rectangular AG-groupoidsâ€, M.Phil. Thesis, University of Malakand, Pakistan, 2017.

B.D. McKay and I. M. Wanless, “On the number of Latin squaresâ€, Ann. Combin., vol. 9, pp. 335-344, 2005.




How to Cite

I. Ahmad, A. Rauf, and M. Khan, “A Comparative Study of Anti-rectangular AG-groupoids and Commutative Semigroups”, The Nucleus, vol. 57, no. 1, pp. 21–26, Sep. 2020.




Most read articles by the same author(s)

1 2 > >>