Involution Abel-Grassmann’s groups and filter theory of Abel-Grassmann’s groups. (English) Zbl 1425.20032
Summary: In this paper, some basic properties and structure characterizations of AG-groups are further studied. First, some examples of infinite AG-groups are given, and weak commutative, alternative and quasi-cancellative AG-groups are discussed. Second, two new concepts of involution AG-group and generalized involution AG-group are proposed, the relationships among (generalized) involution AG-groups, commutative groups and AG-groups are investigated, and the structure theorems of (generalized) involution AG-groups are proved. Third, the notion of filter of an AG-group is introduced, the congruence relation is constructed from arbitrary filter, and the corresponding quotient structure and homomorphism theorems are established.
MSC:
| 20N02 | Sets with a single binary operation (groupoids) |
| 20E34 | General structure theorems for groups |
References:
| [1] | Hall, G.G.; ; Applied Group Theory: New York, NY, USA 1967; . · Zbl 0166.29302 |
| [2] | Howie, J.M.; ; Fundamentals of Semigroup Theory: Oxford, UK 1995; . · Zbl 0835.20077 |
| [3] | Lawson, M.V.; ; Inverse Semigroups: The Theory of Partial Symmetries: Singapore 1998; . · Zbl 1079.20505 |
| [4] | Akinmoyewa, T.J.; A study of some properties of generalized groups; Octogon Math. Mag.: 2009; Volume 7 ,599-626. |
| [5] | Smarandache, F.; Ali, M.; Neutrosophic triplet group; Neural Comput. Appl.: 2018; Volume 29 ,595-601. |
| [6] | Kazim, M.A.; Naseeruddin, M.; On almost semigroups; Port. Math.: 1977; Volume 36 ,41-47. · Zbl 0498.20053 |
| [7] | Kleinfeld, M.H.; Rings with x(yz) = z(yx); Commun. Algebra: 1978; Volume 6 ,1369-1373. · Zbl 0376.17001 |
| [8] | Mushtaq, Q.; Iqbal, Q.; Decomposition of a locally associative LA-semigroup; Semigroup Forum: 1990; Volume 41 ,155-164. · Zbl 0682.20049 |
| [9] | Ahmad, I.; Rashad, M.; Constructions of some algebraic structures from each other; Int. Math. Forum: 2012; Volume 7 ,2759-2766. · Zbl 1257.20064 |
| [10] | Ali, A.; Shah, M.; Ahmad, I.; On quasi-cancellativity of AG-groupoids; Int. J. Contemp. Math. Sci.: 2012; Volume 7 ,2065-2070. · Zbl 1256.20061 |
| [11] | Dudek, W.A.; Gigon, R.S.; Completely inverse AG∗∗-groupoids; Semigroup Forum: 2013; Volume 87 ,201-229. · Zbl 1284.20072 |
| [12] | Zhang, X.H.; Wu, X.Y.; Mao, X.Y.; Smarandache, F.; Park, C.; On neutrosophic extended triplet groups (loops) and Abel-Grassmann’s groupoids (AG-groupoids); J. Intell. Fuzzy Syst.: 2019; . |
| [13] | Wu, X.Y.; Zhang, X.H.; The decomposition theorems of AG-neutrosophic extended triplet loops and strong AG-(l,l)-loops; Mathematics: 2019; Volume 7 . |
| [14] | Kamran, M.S.; Conditions for LA-Semigroups to Resemble Associative Structures; Ph.D. Thesis: Islamabad, Pakistan 1993; . |
| [15] | Shah, M.; Ali, A.; Some structure properties of AG-groups; Int. Math. Forum: 2011; Volume 6 ,1661-1667. · Zbl 1250.20057 |
| [16] | Protić, P.V.; Some remarks on Abel-Grassmann’s groups; Quasigroups Relat. Syst.: 2012; Volume 20 ,267-274. · Zbl 1278.20083 |
| [17] | Mushtaq, Q.; Abelian groups defined by LA-semigroups; Stud. Sci. Math. Hung.: 1983; Volume 18 ,427-428. · Zbl 0488.20055 |
| [18] | Ma, Y.C.; Zhang, X.H.; Yang, X.F.; Zhou, X.; Generalized neutrosophic extended triplet group; Symmetry: 2019; Volume 11 . |
| [19] | Zhang, X.H.; Bo, C.X.; Smarandache, F.; Park, C.; New operations of totally dependent-neutrosophic sets and totally dependent-neutrosophic soft sets; Symmetry: 2018; Volume 10 . |
| [20] | Zhang, X.H.; Mao, X.Y.; Wu, Y.T.; Zhai, X.H.; Neutrosophic filters in pseudo-BCI algebras; Int. J. Uncertain. Quan.: 2018; Volume 8 ,511-526. |
| [21] | Zhang, X.H.; Borzooei, R.A.; Jun, Y.B.; Q-filters of quantum B-algebras and basic implication algebras; Symmetry: 2018; Volume 10 . · Zbl 1437.03180 |
| [22] | Zhan, J.M.; Sun, B.Z.; Alcantud, J.C.R.; Covering based multigranulation (I, T)-fuzzy rough set models and applications in multi-attribute group decision-making; Inf. Sci.: 2019; Volume 476 ,290-318. · Zbl 1442.68236 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
