On Further Study of CA-AG-groupoids
On Further Study of CA-AG-groupoids
Keywords:
AG-groupoid, CA-AG-groupoid, bi-commutative, Stein AG-groupoids, direct productAbstract
A groupoid satisfying the “left invertive law” is called an Abel-Grassmann’s groupoid (or simply AGgroupoid [2]). In literature different names like “left almost semigroup” (LA-semigroup) [3], left invertive groupoid [4] and right modular groupoid [5] has been used by different authors for the said structure. Many properties of AG-groupoids have been studied in [6, 7]. Various aspects of AG-groupoids have been studied in [2, 8 − 14]. In [1,15], we introduced CA-AG-groupoid as a new subclass of AGgroupoid and studies some fundamental properties of it. In the same paper we introduced CA-test for the verification of cyclic associativity for an arbitrary AG-groupoid. We also enumerated CA-AG-groupoids up to order 6 and further classified it into different subclasses.
References
Iqbal, M., I. Ahmad, M. Shah & M.I. Ali. On Cyclic Associative Abel-Grassmann Groupoids. British J. Math and comp. Sci. 12(5): 1 − 16, Article no. BJMCS.21867 (2016).
Protic, P.V. & N. Stevanovic. On Abel-Grassmann’s groupoids (review). Proceeding of Mathematics Conference in Pristina 31 − 38 (1994).
Kazim, M.A. & M. Naseeruddin. On almost semigroups. Portugaliae Mathematica 2: 1 − 7 (1972).
Holgate, P. Groupoids satisfying a simple invertive law. Mathematics Student. 61: 101 − 106 (1992).
Cho, J.R. Pusan, J. Jezek & T. Kepka. Paramedial groupoids. Czechoslovak Mathematical Journal. 49(124): 277 − 290 (1996).
Mushtaq, Q. & S.M. Yusuf. On LA-Semigroups. The Aligarh Bulitun Mathematics 65 − 70: 8 (1978).
Mushtaq, Q. and S.M. Yusuf. On locally associative LA-Semigroups. J. Nat. Sci. Maths. ???????????????????????? (1), 57 − 62: 19 (1979).
Mushtaq, Q. & M.S. Kamran. On LA-semigroups with weak associative law. Scientific Khyber 1(11): 69 − 71 (1989).
Protic, P.V. & M. Bozinovic. Some congruences on an AG**-groupoid. Algebra, Logic and Discrete Math. 879 − 886: 14 − 16 (1995).
Shah, M. I. Ahmad & A. Ali. Discovery of new classes of AG-groupoids. Res. J. Recent Sci. 1(11): 47 − 49 (2012).
Rashad, M. Amanullah, I. Ahmad, & M. Shah. On relations between right alternative and nuclear square AGgroupoids. International Mathematical Forum. Vol. 8(5): 237 − 243(2013).
Stevanovic, N. & P.V. Protic. Some decomposition on Abel-Grassmann’s groupoids. PU. M. A. 355 − 366: 8 (1997).
Mushtaq, Q. & M. Khan. Direct product of Abel Grassmann’s groupoids. Journal of Interdisciplinary Mathematics No. 4, 461 − 467: 11 (2008).
Shah, M. Shah, T. & A. Ali. On the Cancellativity of AG-groupoids. International Mathematical Forum, Vol. 6, no. 44: 2187 − 2194 (2011).
Iqbal, M. The investigation of cyclic associativity in AG-groupoids. M.Phil thesis, Malakand University, Pakistan, Submitted to HEC arxive, (2014).
Jezek, J. & T. Kepka. Medial Groupoids. Academia Nakladatelstvi Ceskoslovenske Akademie Ved. (1983).
Shah, M. A theoretical and computational investigation of AG-groups. PhD thesis, Quaid-i-Azam University Islamabad, Pakistan (2012).
Rashad, M. Investigation and classification of new subclasses of AG-groupoids. PhD thesis, Malakand University, Pakistan (2015).
Aziz-ul-Hakim. Relationship between self-dual AG-groupoid and AG-groupoids. M.Phil. thesis, Malakand University, Pakistan (2014).
