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Paper   IPM / M / 56
School of Mathematics
  Title:   Duality in a generalized model for translation invariant systems
  Author(s):  A. Daneshgar
  Status:   Published
  Journal: Fuzzy Sets and Systems
  Vol.:  83
  Year:  1996
  Pages:   347-352
  Supported by:  IPM
In a previous paper we introduced a generalized model for translation invariant (TI) operators. In this model 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. Also, in that paper, we proved strong reconstruction theorem to show the consistency of this model. This theorem states that for an order-preserving TI operator Y one can explicitly compute Y(A), for any A, from a specific subset of Φ called the base of Y.
    In this paper duality is considered in the same general framework, and in this regard, continuous TI operators are studied. This kind of operators are characterized in terms of critical points of their kernels. A critical point is defined to be a point at which there exists a net which converges to −∞, when −∞ is the value of a nontrivial limit-function of the kernel at that point. A critical point is called to be of first type if for every such net there exists some other point at which this net converges to +∞, and it is called to be of second type otherwise. As the main result of this paper we prove that a well-defined isotone TI operator is continuous if and only if it has no critical point of second type. Moreover, in this case a TI operator has a continuous extension by duality.

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