N‐topo nilpotency in fuzzy neighborhood rings

T M G Ahsanullah,Fawzi Al-Thukair

doi:10.1155/s0161171204305247

T M G Ahsanullah, Fawzi Al-Thukair

Open Access

https://doi.org/10.1155/s0161171204305247

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Abstract

We introduce the notion of N‐topo nilpotent fuzzy set in a fuzzy neighborhood ring and develop some fundamental results. Here we show that a fuzzy neighborhood ring is locally inversely bounded if and only if for all 0 < α < 1, the α‐level topological rings are locally inversely bounded. This leads us to prove a characterization theorem which says that if a fuzzy neighborhood ring on a division ring is Wuyts‐Lowen WNT2 and locally inversely bounded, then the fuzzy neighborhood ring is a fuzzy neighborhood division ring. We also present another characterization theorem which says that a fuzzy neighborhood ring on a division ring is a fuzzy neighborhood division ring if the fuzzy neighborhood ring contains an N‐topo nilpotent fuzzy neighborhood of zero.

Highlights

This paper is a continuation into the investigation of the compatibility of the Lowen fuzzy neighborhood topologies with algebraic structures
We introduce and study the notion of N-topo nilpotent fuzzy set in a fuzzy neighborhood ring
We prove that the notion of bounded fuzzy set introduced in [5] is a good extension

Summary

Introduction

This paper is a continuation into the investigation of the compatibility of the Lowen fuzzy neighborhood topologies with algebraic structures. 3. N-topo nilpotent fuzzy set in a fuzzy neighborhood ring The notion of N-topo nilpotency of an element in a fuzzy neighborhood ring is a good extension of its classical counterpart, which can be seen from Theorem 3.3 below. Let (R, +, ·, t(ΣR)) be a fuzzy neighborhood ring and let μ ∈ IR be left bounded fuzzy set.

Results

Conclusion