In this paper poset valued intuitionistic fuzzy sets are introduced and investigated (for posets in general). It is proved that in case of arbitrary finite posets, represented as a poset of a convenient distributive lattice, a natural version of the definition is obtained, using join-irreducible elements. Two families of cuts and their properties are investigated. Necessary and sufficient conditions under which two families of subsets are families of cuts of a poset valued intuitionistic fuzzy set are given.
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