An extension of overlap functions on convolution lattices

Abstract

Up to now, overlap functions have been expanded into various domains, becoming a significant research topic. The study of extended aggregation operations within a lattice-theoretic structure has garnered momentous interest. In this paper, applying Zadeh's extension principle, we extend overlap functions to convolution lattices. More specifically, (i) we introduce three concepts of extended overlap functions, namely Z-quasi-overlap functions, Z-overlap functions, and $C{E}_L$-Z-overlap functions; (ii) we present some useful properties of extended overlap functions on the set of lattice functions; (iii) we apply these properties to show that an extended overlap function is a $0_{\delta }$-Z-overlap function on convolution lattices, providing domain lattice is complete and co-domain lattice a frame. Last but not least, we anticipate that these findings will apply to the general lattice-theoretic framework for type-2 fuzzy systems, and to various areas of soft computing that involve fuzzy logic connectives in type-2 fuzzy sets.