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.