(U,N)-implications On Bounded Lattices And Partial Orders Induced By T(S)_-uniform Modules

Zadeh proposed the concept of fuzzy sets in 1965,which represented the birth of fuzzy logics and fuzzy mathematics.Since fuzzy logic connectives(such as conjunctions,disjunctions,negations,implications,etc.)play an important role in the theories and applications of fuzzy logics and the reasonable combinations of different logical connectives determine different logical systems,many different types of fuzzy logic connectives have been discussed by scholars on unit intervals or bounded lattices,especially those fuzzy implications which are generated by the existing fuzzy logic connectives,such as(S,N)-implications,(U,N)-implications,R-implications and so on.In recent years,many scholars defined different partial orders by fuzzy logic connectives and further introduced equivalent relations by these partial orders.Then they discussed the equivalence classes which are obtained by equivalent relations.In this paper,the concept of(U,N)-implication on a bounded lattice is introduced and some basic properties of(U,N)-implications on bounded lattices are discussed.Then we give the largest and the smallest disjunctive uninorms on bounded lattices with the co-prime 1,respectively.On the basis of these results,we obtain the largest and the smallest(U,N)-implications on bounded lattices with the co-prime 1,respectively.Finally,we define six partial orders by T-uninorms and S-uninorms on[0,1]and show that these partial orders are irrelevant in general by some examples.The main contents are arranged as follows:Chapter One:Preliminaries.Some basic concepts and relevant properties about prime elements,co-prime elements,uninorms and fuzzy implications on a bounded lattice,T-uninorms and S-uninorms on a unit interval are given.Chapter Two:(U,N)-implications on bounded lattices.Firstly,the concept and some basic properties of(U,N)-implications are given.Then we construct the largest and the smallest disjunctive uninorms on bounded lattices with the co-prime 1.Finally,based on these results,we obtain the largest and the smallest(U,N)implications on bounded lattices with the co-prime 1.Chapter Three:Partial orders induced by T-uninorms and S-uninorms.First of all,the concepts of partial orders ?F,?F' and ?F induced by T-uninorms are given on a unit interval and the relationships between the partial orders and ?are discussed,respectively.Some examples are given to show that the three partial orders induced by T-uninorms are different.Then the concepts of partial orders?H,?H' and ?H induced by S-uninorms are given on a unit interval and the relationships between the partial orders and<are discussed,respectively.Some examples are given to show that the three partial orders induced by S-uninorms are different.Finally,some examples are given to illustrate that the partial orders induced by T-uninorms and S-uninorms are different.