The distributive laws of fuzzy implications over overlap and grouping functions

Overlap and grouping functions, as two kinds of special binary aggregation functions, have been investigated for applications in image processing, classification problems and decision making based on fuzzy preference relations. In addition, after the distributive laws related to fuzzy implications and some special binary aggregation functions have been proposed, there arise many discussions on this research topic. In this paper, we continue studying this topic and discuss the four basic distributive laws of fuzzy implications over overlap and grouping functions. At first, we investigate the four basic distributive laws of fuzzy implications over overlap and grouping functions for the residual implications derived from overlap functions, (G, N)-implications obtained from grouping functions G and fuzzy negations N, and QL-operations derived from overlap functions, grouping functions and fuzzy negations, respectively. And then, we extend the related results when we take into account the residual implications, (G, N)-implications or QL-operations in the four basic distributive laws of fuzzy implications over overlap and grouping functions to arbitrary fuzzy implications satisfying some desirable algebraic properties in the preceding four basic distributive laws.