Next Article in Journal
Neutrosophic Triangular Norms and Their Derived Residuated Lattices
Previous Article in Journal
A Bidirectional Searching Strategy to Improve Data Quality Based on K-Nearest Neighbor Approach
Open AccessArticle

On Two Conjectures of Abel Grassmann’s Groupoids

1
School of Arts and Sciences, Shaanxi University of Science & Technology, Xi’an 710021, China
2
School of Science, Xi’an Polytechnic University, Xi’an 710048, China
*
Author to whom correspondence should be addressed.
Symmetry 2019, 11(6), 816; https://doi.org/10.3390/sym11060816
Received: 13 May 2019 / Revised: 13 June 2019 / Accepted: 16 June 2019 / Published: 20 June 2019
  |  
PDF [216 KB, uploaded 20 June 2019]

Abstract

The quasi-cancellativity of Abel Grassmann‘s groupoids (AG-groupoids) are discussed and two conjectures are partially solved. First, the following conjecture is proved to be true: every AG-3-band is quasi-cancellative. Moreover, a new notion of AG-(4,1)-band is proposed, and it is also proved that every AG-(4,1)-band is quasi-cancellative. Second, the notions of left (right) quasi-cancellative AG-groupoids and power-cancellative AG-groupoids are proposed, and the following results are obtained: for an AG*-groupoid or AG**-groupoid, it is left quasi-cancellative if and only if it is right quasi-cancellative; for a power-cancellative and locally power-associative AG-groupoid, it is left quasi-cancellative if and only if it is right quasi-cancellative. Finally, a general result is proved, that for any AG-groupoid, if it is left quasi-cancellative then it is right quasi-cancellative. View Full-Text
Keywords: Abel-Grassmann’s groupoid (AG-groupoid); AG-3-band; AG*-groupoid; quasi- cancellative; power-cancellative Abel-Grassmann’s groupoid (AG-groupoid); AG-3-band; AG*-groupoid; quasi- cancellative; power-cancellative
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
SciFeed

Share & Cite This Article

MDPI and ACS Style

Zhang, X.; Ma, Y.; Yu, P. On Two Conjectures of Abel Grassmann’s Groupoids. Symmetry 2019, 11, 816.

Show more citation formats Show less citations formats

Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.

Related Articles

Article Metrics

Article Access Statistics

1

Comments

[Return to top]
Symmetry EISSN 2073-8994 Published by MDPI AG, Basel, Switzerland RSS E-Mail Table of Contents Alert
Back to Top