This paper focuses on a generalized definition of fuzzy subsemigroup (ideal) on semigroup. Let L be a completely distributive lattice; we introduce the definition of L -fuzzy ideal and also the novel concept of subsemigroup (ideal) on semigroup. Then, we discuss the necessary and sufficient conditions of L -fuzzy subsemigroup (ideal) measure using the four level cuts of an L -fuzzy set. Moreover, we study the properties of L -fuzzy subsemigroup (ideal) measure. As an application of L -fuzzy subsemigroup (ideal) measure, we obtain the L -fuzzy convexities on a semigroup and bijective semigroup homomorphic mapping is an L -fuzzy isomorphism.