<p>In recent work, the concepts of overlap and grouping functions on complete lattices have been introduced and their related basic algebraic properties have been investigated as well. The present paper can be seen as a continued work along with this research line. More precisely, we focus on an interesting generalization, called general overlap and grouping functions, and provide some constructive methods of these functions via meet and join generators. Here first we introduce the notions of general overlap and grouping functions on complete lattices, and explore the extension of a general overlap function on the subinterval of a complete lattice. In particular, we provide some sufficient and necessary conditions under which the extension of a weaker general overlap function is also a general overlap function of the same form. Subsequently, we introduce the concept of lattice generators, including meet and join generators, of general overlap and grouping functions on the complete lattices, and provide the necessary and sufficient conditions of two one-place functions to be a lattice generators, discussing the relationship between lattice generators of general overlap and grouping functions.</p>