About this Journal Submit a Manuscript Table of Contents
Journal of Mathematics
Volume 2013 (2013), Article ID 475190, 5 pages
http://dx.doi.org/10.1155/2013/475190
Research Article

On Minimal Fuzzy Ideals of Semigroups

1Department of Mathematics, COMSATS Institute of Information Technology, Abbottabad 22060, Pakistan
2Department of Applied Mathematics, Xi'an University of Posts and Telecommunications, Xi'an 710121, China
3School of Chemical & Materials Engineering (SCME), National University of Sciences & Technology, H-12, Islamabad 44000, Pakistan

Received 18 September 2012; Accepted 27 September 2012

Academic Editor: Baoding Liu

Copyright © 2013 Madad Khan et al. 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.

Abstract

The present paper contains the sufficient condition of a fuzzy semigroup to be a fuzzy group using fuzzy points. The existence of a fuzzy kernel in semigroup is explored. It has been shown that every fuzzy ideal of a semigroup contains every minimal fuzzy left and every minimal fuzzy right ideal of semigroup. The fuzzy kernel is the class sum of minimal fuzzy left (right) ideals of a semigroup. Every fuzzy left ideal of a fuzzy kernel is also a fuzzy left ideal of a semigroup. It has been shown that the product of minimal fuzzy left ideal and minimal fuzzy right ideal of a semigroup forms a group. The representation of minimal fuzzy left (right) ideals and also the representation of intersection of minimal fuzzy left ideal and minimal fuzzy right ideal are shown. The fuzzy kernel of a semigroup is basically the class sum of all the minimal fuzzy left (right) ideals of a semigroup. Finally the sufficient condition of fuzzy kernel to be completely simple semigroup has been proved.