A note on self-dual AG-groupoids

AZIZ-UL-HAKIM; I. AHMAD; M. SHAH

Author(s)

AZIZ-UL-HAKIM Department of Mathematics, University of Malakand, Chakdara Dir(L), Pakistan.

I. AHMAD Department of Mathematics, University of Malakand, Chakdara Dir(L), Pakistan.

M. SHAH Department of Mathematics, Government Post Graduate College Mardan, Pakistan.

Abstract

In this paper, we enumerate self-dual AG-groupoids up to order 6; and classify them on the basis of commutativity and associativity. A self-dual AG-groupoid-test is introduced to check an arbitrary AG-groupoid for a self-dual AG-groupoid. We also respond to an open problem regarding cancellativity of an element in an AG-groupoid. Some features of ideals in self-dual AG-groupoids are explored. Some desired algebraic structures are constructed from the known ones subject to certain conditions and some subclasses of self-dual AG-groupoids are introduced.

Publication Details

Page(s) 1-15

DOI DOI not available

Published Journal: Journal of Prime Research in Mathematics, Volume: 12, Issue: 1, Year: 2016

Request the full text from PASTIC

Keywords

AG-groupoid, Enumeration, , AMS SUBJECT, connected set, 20N05, ideal, 20M12, anti-rectangular AG-groupoid, Primary 20N02

References

I. Ahmad ,Shah M.,Fuzzy M. .2012 .. Life Science Journal, 9(4) : 3931-3936.

Amanullah I.,Ahmad M.,Shah M. .2013 .On the Equal-height Elements of Fuzzy AGsubgroups. Science Journal, 10(4) : 3143-3146.

Amanullah M.,Rashad I. Ahmad .2016 .Abel-Grassmann's Groupoids of Modulo Matrices. , 35(1) : 63-70.

Amanullah M.,Rashad M.,Shah M. .2014 .On Modulo AG-groupoids. Journal of Advances in Mathematics 8, (3) : 1606-1613.

Hakim .2014 .Relationship between self-dual AG-groupoid and AG-groupoids, M.Phil thesis. , : .

Celakoska-Jordanova V.,Janeva B. .2009 .Free left commutative groupoids. , : 687-694.

Cho J. R.,Kepka T.,Praha T. .1996 .. Czechoslovak Mathematical Journal, 49(124) : .

Distler A.,Shah M.,Sorge V. .2011 .. Enumeration of AG-groupoids, Lecture Notes in Computer Science, 6824 : 1-14.

Kazim M. A.,Naseerudin M. .1977 .. Portugaliae Mathematica, 36(1) : .

Khan M.,Mushtaq Q. .2003 .Ideals in left almost semigroups. In proceedings of 4th International Pure Mathematics Conference, : 65-77.

Mushtaq Q. .2004 .Zeroids and Idempoids in AG-groupoids. Quasigroups and related Systems, 11 : 79-84.

Mushtaq Q.,Iqbal Q. .1990 .Decomposition of a locally associative LA-semigroup. Semigroup Forum, 41 : 155-164.

Mushtaq Q.,Khan M. .2006 .On left almost ring. In proceedings of 7th International Pure Mathematics Conference, : .

Mushtaq Q.,Khan M. .2007 .A note on an Abel-Grassmann's 3-band. Quasigroups and Related Systems, 15 : 295-301.

Mushtaq Q.,M. Q. .2013 .Khan and Kar Ping Shum, Topological structures on LA-Semigroups. Bull. Malays. Math. Sci. Soc, 2(4) : 901-906.

Protic P. V.,Bozinnovic M. .1995 .-groupoid. Algebra Logic and Discrete Mathematics, Nis, 3(9) : 879-886.

Protic P. V.,Stevanovic N. . .AG-test and some general properties of AbelGrassmann's groupoids, PU. M. A. No. 4, 6(1995) : 371-383.

Rashad M.,I. Ahmad M.,Shah M. .2016 .Some general properties of Stein AG-groupoids and Stein AG-test. Sindh Univ. Res. Jour. (Sci. Ser, 48(3) : 679-684.

Shah M. .2012 .A theoretical and computational investigation of AG-groups. PhD thesis, : .

Shah M.,I. M. .2012 .Ahmad and A. Ali, Discovery of new classes of AG-groupoids. Research Journal of Recent, 1(11) : 47-49.

Shah M.,Ahmad I.,Rashad M. .2013 .On Introduction of New Classes of AG-groupoids. Research Journal of Recent, 2(1) : 67-70.

Shah T.,I. Rehman T. .2010 .On gamma-ideals and gamma-bi-ideals in gamma- AGgroupoids. International Journal of Algebra, 4(6) : 267-276.

Stevanovic N.,Protic P.V. .1999 .In°ations of the AG-groupoids, Novi Sad J. , 29(1) : 19-26.