Constructing Θ-Ξ functions via multiplicative generators on a completely distributive lattice

Abstract $\varTheta $ -$\varXi $ functions are the unified form of 0-overlap and 1-grouping functions. And the unified form of $0$-overlap and $1$-grouping functions via multiplicative generators on unit interval has been researched. In fact, either real number sets or interval-valued sets are special cases of posets or lattices and many questions in applications involve incomparable elements. Based of above questions, we continue to consider this topic and construct the unified form of $0$-overlap and $1$-grouping functions via multiplicative generators on a completely distributive lattice. Finally, we investigate some properties including extended migrativity and homogeneity of $\varTheta $-$\varXi $ functions constructed by generalized multiplicative generators triples.