On Fuzzy Deductive Systems of Hilbert Algebras

Abstract

In this paper, we study fuzzy deductive systems of Hilbert algebras whose truth values are in a complete lattice satisfying the infinite meet distributive law. Several characterizations are obtained for fuzzy deductive systems generated by a fuzzy set. It is also proved that the class of all fuzzy deductive systems of a Hilbert algebra forms an algebraic closure fuzzy set system. Furthermore, we obtain a lattice isomorphism between the class of fuzzy deductive systems and the class of fuzzy congruence relations in the variety of Hilbert algebras.

The Fuzzy Prime Spectrum of Partially Ordered Sets

“…Te space of fuzzy prime ideals, called the hull-kernel topology for fuzzy prime ideals, has been studied by many authors in diferent classes of general algebraic structures such as rings (see [14]), modules (see [15]), hemirings (see [16]), semirings and Γ-Semirings (see [17]), distributive lattices (see [18]), C-algebras (see [19]), etc. Tis topology provides a powerful tool to explore and hence to utilize the important topological properties that the class of fuzzy prime ideals possesses inherently.…”