Fuzzy logic is a many-valued logical system with truth values within the subset[0, 1] of real numbers. The set[0, 1] is a totally ordered set, and thus it has limitations in application to various real-life examples. Therefore, generalized many-valued logical systems that can be applied to more various examples of real life need to be studied and their properties investigated. In this paper, we introduce the definitions and properties of lattice implication algebras, Heyting semi-lattices, and DBCK-algebras as algebraic structures that can generalize existing fuzzy logic. We also investigate the fuzzy implications defined in the set[0, 1] and their properties, and propose algebraic structures in which fuzzy logic with such implications can be generalized. To do this, we find out the necessary and sufficient conditions for an algebraic structure[L, →, ] to be DBCK-algebra. Also, we prove that lattice implication algebras and Heyting semilattices are DBCK-algebras, and that the algebraic structure, which is a lattice implication and a Heyting semilattice, is an equivalent notion to Boolean algebra.