Fuzzy Equivalences Based On (S,n)-implications And QL-implications

In this thesis, we deal with fuzzy equivalences based on fuzzy implications. The main research includes general properties of fuzzy equivalences based on fuzzy implications and the detailed investigation into fuzzy equivalences based on (S,n)-implications and QL-implications under certain conditions.Firstly, we give a brief introduction to some basic concepts of fuzzy logic connectives and related results involved in the thesis. Then properties of three classes of usual fuzzy implications (R-implication,(S,n)-implication and QL-implication) in the literature are recalled, which constitute the basis of our investigation into fuzzy equivalences.Next, we start our discussion based on the results on classic equivalences and the work done by Belohlave on R-implications. We first give general properties of implication-based fuzzy equivalences and prove that the some classic results can be extended to the fuzzy case without any conditions. Afterwards, as a complementary to the Belohlave’s research, we carry out a further study on R-implication-based fuzzy equivalences. Under the condition that T is left continuous, we show that some results of classic equivalences can be extended to R-implication-based fuzzy equivalences. Then we discuss properties of (S,n)-implication-based fuzzy equivalences in detail. When n is a strong negation, by using the contrapositive symmetry of (S, n)-implications, we present a fuzzy version of the classic contrapositive symmetry. Then, by employing the condition(C)proposed by Peng etc., we prove the T-transitivity of (S, n)-implications with which the T-transitivity of (S,n)-implication-based equivalences is verified. Under the same condition, we also discuss other properties of fuzzy equivalences based on (S,n)-implications. As a result, many results of classic equivalences, are extended to this kind of fuzzy equivalences. Finally, we turn to fuzzy equivalences based on QL-implications and generalize a few of properties of classic equivalences such as T-transitivity etc. in the presence of the condition (C). In view that the condition (C) is frequently used in our investigation, we supply some counterexamples to illustrate that the involved results are not necessarily true if the condition is absent.To sum up, through our research, fuzzy equivalences based on (S,n)-implications and QL-implications are better understood, and thus the theoretic research of fuzzy equivalences is enriched.