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Characterization of fuzzy Minkowski algebra

There are various fuzzy morphologies (Minkowski algebras), these depending on the particular fuzzifification of set inclusion that is employed for the definition of erosion. Set-inclusion fuzzification depends upon the choice of an indicator for set inclusion and, based upon a collection of nine axioms, a class of indicators results such that each indicator in the class yields a Minkowski algebra in which a certain core of the ordinary propositions typically associated with mathematical morphology are valid. By going a bit further and postulating a certain mathematical form for the indicator, one obtains fitting characterizations for the basic operators. In ordinary crisp-set binary morphology, certain fundamental representation theorems hold, specifically the Matheron representations for increasing, translation invariant mappings and for T-openings. The definition of a T-opening extends for fuzzy T-openings. There is also a weakened version of the Matheron kernel representation for increasing, translation invariant mappings.