Overlap and grouping functions on complete lattices

Abstract

Recently, Paiva et al. introduced the concepts of lattice-valued overlap and quasi-overlap functions, and showed the migrativity, homogeneity and other properties of (quasi-) overlap functions on bounded lattices. In this paper, we continue to consider this research topic and study overlap and grouping functions on complete lattices in order to extend the continuity of these two operators from the unit closed interval to the lattices status by using join-preserving and meet-preserving properties of binary operators on complete lattices. More precisely, firstly, we introduce the notion of overlap functions on complete lattices and give two construction methods of them. Secondly, we show some basic properties of overlap functions on complete lattices. In particular, we introduce the concept of (∧,∨)-combination of overlap functions and extend the notions of migrativity and homogeneity of overlap functions on bounded lattices to the so-called (α,B,C)-migrativity and (B,C)-homogeneity of overlap functions on complete lattices, respectively, where α belongs to the complete lattice and B and C are two binary operators on the complete lattice, and then we focus on these properties along with the cancellation law of overlap functions on complete lattices. Finally, we give an analogous discussion for grouping functions on complete lattices.