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“School of Mathematics”

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Paper   IPM / M / 58
School of Mathematics
  Title:   Residuated semigroups and morphological aspects of translation invariant systems
  Author(s):  A. Daneshgar
  Status:   Published
  Journal: Fuzzy Sets and Systems
  Vol.:  90
  Year:  1997
  Pages:   69-81
  Supported by:  IPM
The main goal of this paper is to verify classical properties of morphological operators within the general model of translation invariant (TI) systems. In this model, TI operators are defined on the space of LG-fuzzy sets Φ, i.e. Φ = {A:G →Ω∪{−∞}}, in which G is an abelian group and Ω is a complete lattice ordered group. A TI operator Y is an operator on Φ which is invariant under translation on G and Ω as groups. We consider the generalization of Minkowski addition ⊕ on Φ and we emphasize that (Φ, ⊕) is an involutive residuated topological monoid. We verify all properties of traditional (set-theoretic) morphological operators as well as classical representations (Matheron, 1967) for openings, closures and granulometries in this general setting. We also study spectral (skeleton) decompositions in this model, using the same techniques as in the crisp case (Goutsias and Schonfeld, 1991). This formal approach not only clarifies the role of morphological operators, but also it gives rise to simpler and clearer proofs using standard results of the theory of residuated semigroups and categories.

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