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1 2003

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Shadowed Sets and Related Algebraic Structures

Cattaneo G., Ciucci D.

Fundamenta Informaticae

2003

Vol. 55, nr 3,4

255-284

BZMVdM algebras are introduced as an abstract environment to describe both shadowed and fuzzy sets. This structure is endowed with two unusual complementations: a fuzzy one \lnot and an intuitionistic one ~ . Further, we show how to define in any BZMVdM algebra the Boolean sub-algebra of exact elements and to give a rough approximation of fuzzy elements through a pair of exact elements using an interior and an exterior mapping. Then, we introduce the weaker notion of pre-BZMVdM algebra. This structure still have as models fuzzy and shadowed sets but with respect to a weaker notion of intuitionistic negation ~ a with a Î [0,1/2). In pre-BZMVdM algebras it is still possible to define an interior and an exterior mapping but, in this case, we have to distinguish between open and closed exact elements. Finally, we see how it is possible to define a-cuts and level fuzzy sets in the pre-BZMVdM algebraic context of fuzzy sets.