Shadowed sets and related algebraic structures

BZMV(dM) 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 and an intuitionistic one similar to. Further, we show how to define in any BZMV(dM) 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-BZMV(dM) algebra. This structure still have as models fuzzy and shadowed sets but with respect to a weaker notion of intuitionistic negation similar to(alpha) with alpha is an element of [0, 1/2). In pre-BZMV(dM) 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 alpha-cuts and level fuzzy sets in the pre-BZMV(dM) algebraic context of fuzzy sets.