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In this paper, the notions of an L-sublattice, L-ideal, and L-dual ideal of an L-lattice are introduced. The concepts of an L-maximal ideal and an L-prime ideal in an L-lattice are also defined. The present paper also gives the precise structures of an L-sublattice, L-ideal (dual ideal) generated by an L-subset of an L-lattice. Finally, it is proved that under certain conditions, an L-maximal ideal in an L-lattice is L-prime. It is important to note that, in our studies, the evaluation lattice changes from [0,1] to a lattice L and the parent structure shifts from an ordinary lattice to an L-lattice.