Gr. Moisil [1] introduced into consideration the Dual Intuitionistic Logic. A.V. Kuznetsov [2] discovered the connection of this logic with the theory of distributive lattices. Dual Intuitionistic Logic [1] (DIL) is obtained from Propositional Intuitionistic Logic by enriching it with dual logic operations to implication and negation. Among the possible approaches of a problem in DIL is natural to study at the beginning the concerned problem for the case of intermediate logics between DIL and classical logic that are defined by finite distributive lattices. The simplest of these is the chain logic of three elements, which represents a specific fragment of trivalent general logic and coincides with the Lukasiewici trivalent logic. In the present paper we formulate four criterions of functional completeness in the chain extensions of Dual Intuitionistic Logic and also in the simplest not chain extensions of this logic.