A Comparative Study of AG*-groupoidsy


  • I. Ahmad University of Malakand, Chakdara
  • R. Naz University of Malakand, Chakdara
  • M. Rashad University of Malakand, Chakdara
  • A. Khan University of Malakand, Chakdara



A magma satisfying the identityis called an LA-semigroup or AG-groupoid. We investigate the concept of AG*-groupoid, as a subclass of AG-groupoids and establish some relations of AG*-groupoids with other known subclasses of AG-groupoid, like: Stein, Stein*, Jordan and Schroder AG-groupoids. Furthermore, we use the latest computational techniques of Mace-4 and Prover-9 to provide various examples and counterexamples for our relevant investigation.

Author Biographies

I. Ahmad, University of Malakand, Chakdara

Department of Mathematics

R. Naz, University of Malakand, Chakdara

Department of Mathematics

M. Rashad, University of Malakand, Chakdara

Department of Mathematics

A. Khan, University of Malakand, Chakdara

Department of Computer Science


C. Arthur, "On Jacobi's elliptic functions, in reply to the Rev. Brice Brown in and on quaternions", Philosophical Magazine, vol. 26, pp. 208–211, 1845.

P. Dehornoy, “Braids and Self-distributivity”, Progress in Mathematics, vol. 192, 2000.

H.O. Pflugfelder, “Historical Notes on Loop Theory”, Commentationes Mathematicae Universitatis Carolinae, vol. 41, pp. 359-370, 2000.

P. Nagy and K. Strambach, “Loops in Group Theory and Lie Theory”, deGruyter, Berlin. http://dx.doi.org/10.1515/ 9783110900583.

H. Braun and M. Koecher, “Jordan Algebren (German)”, Springer-Verlag, Berlin and New York. http://dx.doi.org/10.1007/978-3-642-94947-0,1966.

N. Jacobson, “Structure and Representations of Jordan Algebras” American Mathematical Society, Providence, 1968.

K. McCrimmon, “A Taste of Jordan Algebras”, Springer-Verleg, New York, 2004.

M. Kazim and M. Naseeruddin, “On almost semigroups”, Alig. Bull. Math., 2, 1-7, 1972.

J. Cho, Pusan, J. Jezek, and T. Kepk, “Paramedial groupoids”, Czechoslovak Mathematical Journal, vol. 49, no. 124, pp. 277-290, 1996.

P. Protic and N. Stevanovic, “Abel-Grassmann’s bands”, Quasigroups and Related Systems, vol. 11, no. 1, pp. 95-101, 20004.

N. Stevanovic and P. Protic, “Composition of Abel-Grassmann's 3-bands”, Novi Sad J. Math., vol. 34, no. 2, pp. 175-182, 2004.

Amanullah, M. Rashad, I. Ahmad and M. Shah, “On modulo AG-groupoids”, Journal of Advances in Mathematics, vol. 8, no 3, pp. 1606-1613, (2014).

I. Ahmad, Amanullah, M. Shah, “Fuzzy AG-subgroups”, Life Science Journal, vol. 9, no. 4, pp. 3931-3936, 2012.

Amanullah, I. Ahmad and M. Shah, “On the equal-height elements of fuzzy AG-subgroups”, Life Science Journal, vol. 10, no. 4, 3143-3146, 2013.

M. Rashad, Amanullah and I. Ahmad, “Modulo matrix AG-groupoids and modulo AG-groups”, Mehran University Research Journal of Engineering & Technology, vol. 35, no. 1, pp. 63-70, 2016.

M. Shah, “A theoretical and computational investigation of AG-groups”, PhD. thesis, Quaid-i-Azam University Islamabad, Pakistan, 2012.

F. Karaslan, I. Ahmad and Amanullah, “Bipolar soft groups, Journal of Intelligent and Fuzzy Systems”, vol. 31, pp. 651-662, 2016.

Amanullah, I. Ahmad and F. Karaaslan, “Cubic Abel-Grassmann’s Subgroups”, Journal of Computational and Theoretical Nanoscinece, vol. 13, pp. 1-8, 2016.

A. Khan, Y. B. Jun, T. Mahmood, “Generalized fuzzy interior ideals in Abel Grassmann's groupoids”, International Journal of Mathematics and Mathematical Sciences 2010.

J. R. Cho, Pusan, J. Jezek and T. Kepka, “Paramedial groupoids”, Czechoslovak Mathematical Journal, vol. 49, no. 124, pp. 277-290, 1996.

P. Holgate, “Groupoids satisfying a simple invertive law”, Math. Students. vol. 61, pp. 101-106, 1992.

P. V. Protic and N. Stevanovic, “On Abel-grassmann's groupoids (Review)”, Proc. of Mathematics Conf. in Pristina, pp. 31-38, 1994.

I. Ahmad and M. Shah, “Paramedial and Bol* Abel Grasmann’s groupoids”, The Nucleus, vol. 52, no. 3, pp. 130-137, 2015.

I. Ahmad, M. Rashad and M. Shah, “Some properties of AG*-groupoid”, Research Journal of Recent Sciences, vol. 2, no. 4, pp. 91-93, April 2003.

I. Ahmad, M. S. Jabbar and M. Rashad, “A note on Rectangular* AG-groupoids”, Sci. Int., vol. 28, no. 2, pp. 873-878, 2016.

M. Rashad, “Investigation and classification of some new classes of AG-groupoids”, Ph.D. thesis, Department of Mathematics, University of Malakand, Chakdara, Dir (L), 2015.

M. Shah and S. Shpectorov and A. Ali, “AG-groups and other classes of right Bolquasi groups” Available at http://arxiv.org/abs/1106.2981v2, 2011.




How to Cite

I. Ahmad, R. Naz, M. Rashad, and A. Khan, “A Comparative Study of AG*-groupoidsy”, The Nucleus, vol. 53, no. 4, pp. 269–274, Dec. 2016.




Most read articles by the same author(s)