Spectrum of prime \(L\)-fuzzy \(h\)-ideals of a hemiring.
type: Item
instance of: scholarly article
title: Spectrum of prime \(L\)-fuzzy \(h\)-ideals of a hemiring.
zbMATH Open document ID: 1201.16039
author: H. V. Kumbhojkar
published in: Fuzzy Sets and Systems
publication date: 2010-07-28T00:00:00Z
review text: The author redefines the notion of a prime fuzzy \(h\)-ideal of a hemiring so that it is not necessarily 2-valued. Let \(S\) be a commutative hemiring with identity. In this case, a non-constant fuzzy \(h\)-ideal \(P\) of \(S\) is prime if \(\forall a,b\in S\), either \(P(ab)=P(a)\) or \(P(ab)=P(b)\). Let \(X\) denote the set of all prime fuzzy \(h\)-ideals \(P\) of \(S\) with membership values in a fixed lattice \(L\) and such that \(P(0)=1\). The set \(X\) is called the spectrum of prime fuzzy \(h\)-ideals of \(S\). Let \(V(A)=\{P\in X\mid A\subseteq P\}\), where \(A\) is a fuzzy subset of \(S\). For \(a\in S\), let \(V(a)=\{P\in X\mid P(a)=1\}\). It is shown that the set \(\{V(\chi_R)\mid R\subseteq S\}\) forms a system of closed sets for a topology on \(X\), where \(\chi_R\) is the characteristic function of \(R\) in \(S\). Let \(X(a)=X\setminus V(a)\). Then \(X(a)=\{P\in X\setminus P(a)\neq 1\}\). Let \(\mathcal S=\{X(a)\mid a\in S\}\). The author shows that \(S\) is a base for a topology on \(X\). The open subsets of \(X\) are the \(X(\chi_T)\), where \(T\) is any subset of \(X\). The topology is completely determined by the semiprime fuzzy \(h\)-ideals of \(S\). The author shows that \(X\) is compact and preserves homomorphisms between hemirings. The correspondence associating a hemiring with its spectrum of prime fuzzy \(h\)-ideals defines a contravariant functor from the category of commutative hemirings with identity into the category of compact topological spaces. Valuation lattices for all fuzzy subsets are assumed to be complete Heyting algebras.
reviewed by: John N. Mordeson
Mathematics Subject Classification ID: 16Y99
Mathematics Subject Classification ID: 16D25
Mathematics Subject Classification ID: 13A99
Mathematics Subject Classification ID: 03E72
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 54B35
Mathematics Subject Classification ID: 54A40
zbMATH DE Number: 5762248
zbMATH Keywords: hemirings
zbMATH Keywords: prime fuzzy ideals
zbMATH Keywords: semiprime fuzzy ideals
zbMATH Keywords: prime spectra
zbMATH Keywords: compact topological spaces
zbMATH Keywords: contravariant functors
MaRDI profile type: Publication
full work available at URL: j.fss.2009.10.006
OpenAlex ID: W2043231973
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DOI: 10.1016/J.FSS.2009.10.006
description: scientific article; zbMATH DE number 5762248