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IPM
30
YEARS OLD

“School of Mathematics”

Paper   IPM / M / 57
   School of Mathematics
  Title: Thresholding in a generalized model for translation invariant systems
  Author(s): A. Daneshgar
  Status: Published
  Journal: Fuzzy Sets and Systems
  Vol.: 85
  Year: 1997
  Pages: 391-395
  Supported by: IPM
  Abstract:
In a previous paper we introduced a generalized model for translation invariant (TI) operators. We considered the space, Φ, of all maps from an abelian group G to Ω∪{−∞}, called LG-fuzzy sets, where Ω is a complete lattice ordered group; and we defined TI operators on this space. In this paper thresholding is considered in the same general framework, and in this regard, positive TI operators are studied. The threshold of an LG-fuzzy set A at level t is defined to be the set {gG:A(g) ≥ t }. Also, a TI operator is defined to be positive if its kernel consists of elements which are positive except possibly at some points which are −∞. As the main result of this note, it is proved that a positive and isotone TI operator commutes with thresholding if and only if it has a crisp kernel (or base). Also, the theorem is used to derive a similar result for an operator whose kernel is bounded from below.

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