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Open AccessArticle

Involution Abel–Grassmann’s Groups and Filter Theory of Abel–Grassmann’s Groups

Department of Mathematics, Shaanxi University of Science & Technology, Xi’an 710021, China
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Symmetry 2019, 11(4), 553; https://doi.org/10.3390/sym11040553
Received: 12 March 2019 / Revised: 5 April 2019 / Accepted: 8 April 2019 / Published: 17 April 2019
(This article belongs to the Special Issue Discrete Mathematics and Symmetry)
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Abstract

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. View Full-Text
Keywords: Abel–Grassmann’s groupoid (AG-groupoid); Abel–Grassmann’s group (AG-group); involution AG-group; commutative group; filter Abel–Grassmann’s groupoid (AG-groupoid); Abel–Grassmann’s group (AG-group); involution AG-group; commutative group; filter
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Zhang, X.; Wu, X. Involution Abel–Grassmann’s Groups and Filter Theory of Abel–Grassmann’s Groups. Symmetry 2019, 11, 553.

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