Down and up operators associated to fuzzy relations and t-norms: A definition of fuzzy semi-ideals

By means of the use of a t-norm T on a complete lattice L and of a L-fuzzy relation R on a set X, there are stated definitions of LF-semi-ideal and of its dual LF-semi-filter of the set X. It is shown that both the poset of the LF-semi-ideals and one of the LF-semi-filters determined by the respective restrictions of the usual order in the class of L-fuzzy sets, are complete lattices. Another characterization of these concepts, as down and up L-fuzzy sets, is made via properties of the L-fuzzy relation R. At the end, we give some constructive methods with which the set of LF-semi-ideals and the set of down L-fuzzy sets can be determined. Examples and particular cases of the theory are presented.

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