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.
Highlights
Introduction e pioneering work ofZadeh [1] on fuzzy subsets of a set has been extensively applied to many scientific fields. is concept was adapted by Rosenfeld [2] to define fuzzy subgroups of a group
We show that μ is an L-fuzzy deductive system of A
We show that αD(S) is the smallest L-fuzzy deductive system of A containing αS
Summary
We recall some definitions and basic properties of deductive systems of a Hilbert algebra. A nonempty subset D of a Hilbert algebra A is called a deductive system on A if it satisfies the following conditions:. One can check that the lattice of deductive systems of a Hilbert algebra A is closed under arbitrary intersection so that for any subset S of A, always there exists a smallest deductive system D(S) of A containing S. It is called the deductive system of A generated by S. By an L-fuzzy subset μ of A, we mean a mapping μ: A ⟶ L. e set μ(x): x ∈ A is called the image of μ and is denoted by Im(μ). For each y ∈ A and we call it an L-fuzzy point of A
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