L-valued quasi-overlap functions, L-valued overlap index, and Alexandroff’s topology

In one recent work, Paiva et al. generalized the notion of overlap functions to the context of lattices and introduced a weaker definition, called a quasi-overlap, that arises from the removal of the continuity condition. In this article, quasi-overlap functions on lattices are equipped with a topological space structure, namely, Alexandroff’s spaces. Some examples are presented and theorems related to the migrativity and neutral element properties are provided. It is shown that, in these spaces, the concepts of overlap and quasi-overlap functions coincide. Also, the notion of overlap index is extended to the context of L-fuzzy sets. L-valued overlap indices are obtained by adding degrees of quasi-overlap functions on bounded lattice L, as well as quasi-overlap functions are obtained via L-valued overlap indices and some examples are presented. Finally, the concepts of migrativity and convex sum are extended to the context of L-valued overlap index.