Topological Isomorphism Between Fuzzy-Quantifier-Represented Gergonne Relations and Their Diagrams

A Gergonne relation is traditionally understood as one of the 5 operations of the two sets {x} and {y}, viz. Being included, coincidence, inclusion, intersection, and separation. It is commonly used in interpreting categorical propositions with the structure like (Qx R y), wherein, Q is a (traditional) quantifier of either a universal or a particular quantifier, and R is a relation mark commonly interpreted as “be/be not” or “(does not) apply to” to represent an equivalent or its negation of the two relatives x and y. For illustrating the Gergonne relation, the Gergonne diagram is considered as one kind of double Euler circles in the 5 interactions. However, the illustration has been controversial, in particular on the correspondence between the Gergonne diagrams and the proper propositional forms of the Gergonne relations. The author of the present paper has introduced a propositional form (Qx R Qy) to state an expanded Gergonne relation, wherein, Q is a fuzzy quantifier of the universal or partial or existential to avert the imperfections due to the confusion of the mentioned traditional quantifiers, and the ambiguity of losing the binder of y. Correspondingly, diagrams interpreting the expanded Gergonne relation are to be introduced by means of discovering intrinsic topological characteristics of the diagrams. Especially, the isomorphism between the discrete topology product with the 5 operations and the 2-dimensional expanded Gergonne diagrams made up of 5 double Euler circles are proven. The results are expected to turn syllogisms based on the expanded Gergonne relations into topological computations.