On the relationship of extended necessity measures to implication operators on the unit interval

doi:10.1016/0020-0255(94)00046-E

Information Sciences

Volume 82, Issues 3–4, January 1995, Pages 129-145

https://doi.org/10.1016/0020-0255(94)00046-E Get rights and content

Abstract

By an extended necessity measure on a set X we mean a function from I^X to the unit interval I such that the measure of an arbitrary meet of fuzzy subsets of X equals the infimum of their measures. We address the question of whether a given extended necessity measure on X is induced, through a given implication operator

on I, from some (unknown) normalized fuzzy subset ν of X in the following manner: The measure of μ ϵ I^X is the degree of containment of ν in μ, according to the implication

. We provide a simple complete answer to this question in the case

is in one of the following three infinite sets of implication operators: the S-type implications, the residuated implications, and the continuous implications that satisfy the exchange principle.

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On the relationship of extended necessity measures to implication operators on the unit interval

Abstract

Fuzzy Sets and Systems

Fuzzy Sets and Systems

Fuzzy Sets and Systems

Inform. Control