Spectrum of prime \(L\)-fuzzy \(h\)-ideals of a hemiring.
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.
scientific article; zbMATH DE number 5762248