Cartesian product of interval-valued fuzzy ideals in ordered semigroup

Hidayat Ullah Khan
Department of Mathematics, University of Malakand, Khyber Pakhtoonkhwa, Pakistan.
Asghar Khan
Department of Mathematics, Abdul Wali Khan University Mardan, Mardan, Khyber Pakhtoonkhwa, Pakistan.
Nor Haniza Sarmin
Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia,
81310 UTM, Johor Bahru, Johor, Malaysia.

\(^{1}\)Corresponding Author: hidayatullak@yahoo.com

Copyright © 2016 Hidayat Ullah Khan, Asghar Khan, Nor Haniza Sarmin. 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.

Published: December, 2016.

Abstract

Interval-valued fuzzy set theory is a more generalized theory that can deal with real world problems more precisely than ordinary fuzzy set theory. In this paper, the concepts of interval-valued fuzzy (prime, semiprime) ideal and the Cartesian product of interval-valued fuzzy subsets have been introduced. Some interesting results about Cartesian product of interval-valued fuzzy ideals, interval-valued fuzzy prime ideals, intervalvalued fuzzy semiprime ideals, interval-valued fuzzy bi-ideals and intervalvalued fuzzy interior ideals in ordered semigroups are obtained. The purport of this paper is to link ordinary ideals with interval-valued fuzzy ideals by means of level subset of Cartesian product of interval-valued fuzzy subsets.

Keywords:

Interval-valued fuzzy left (right) ideals, Interval-valued fuzzy bi-ideals, Interval-valued fuzzy interior ideals, Interval-valued fuzzy prime ideals, Interval-valued fuzzy semiprime ideals.