On an Application of Fuzzy Relations in Biogeography

Abstract

Starting from a quasi-ordering relation on a set, by a well-known construction, an equivalence relation could be made, as well as an ordering relation on the quotient set. In this paper, we give a similar fuzzy construction with lattice-valued fuzzy relations. If ρ is a (crisp) correspondence from set A to set B, then a quasi-ordering relation θ on the set B could be defined in the following way: (x,y)∈θ, if and only if (∀z∈A)((z,x)∈ρ→(z,y)∈ρ). In this paper, an analogous fuzzy construction is given for a special type of fuzzy correspondences which appear in biogeography. From this correspondence, a quasi-ordering relation is obtained, as well as the corresponding fuzzy similarity relation and fuzzy ordering relation on the set of similarity classes. This construction was developed to be applied in solving some problems of determining connections of different biogeographical areas according to recorded species. Here, those results have been applied to determine connections of biogeographical territories in the Balkan Peninsula, according to different species of Genus Cheilosia Meigen 1822 and related genera (Diptera: Syrphidae) (hover-flies).