On the ordinal sum of fuzzy implications: New results and the distributivity over a class of overlap and grouping functions

Similar to the construction of ordinal sums of overlap functions, Baczynski et al. introduced two new kinds of ordinal sums of fuzzy implications without additional restrictions on summands in 2017. In this paper, based on the ordinal sum of fuzzy implications with the first method, we discuss its basic properties, such as iterative Boolean law, right ordering property and strong boundary condition. Meanwhile, we give characterizations of such ordinal sum of fuzzy implications that is a QL-implication constructed from tuples (O, G, NT) or a D-implication derived from grouping function G, and show some conclusions about its relations with (G, N)- and RO-implications. Moreover, we study the distributivity of such ordinal sum of fuzzy implications over a class of overlap and grouping functions. More specifically, we give necessary and sufficient conditions under which this ordinal sum of fuzzy implications is distributive over overlap and grouping functions satisfying some conditions.(c) 2021 Elsevier B.V. All rights reserved.