T-partial order and its properties

The main aim of this thesis is to investigate properties of T-partial order, denoted by ?_T, defined on a bounded lattice L, to define on the set of all incomparable elements with arbitrary but fixed c?[0,1] according to ?_T for any t-norm T on [0,1] to study on this set and to construct a t-norm T such that ([0,1],?_T ) is a lattice. This study consists of two main chapters. In Chapter 1, some definitions and theorems which are crucial for our study are stated. Chapter 2 contains four parts. In the first part, it is defined that the notions order-strong and order-weak for t-norms on a bounded lattice L and it is determined the order-weakest and the order strongest t-norms. In the second part, it is defined that an equivalence relation ß on the class of t-norms on [0,1] and it is investigated properties of this relation. In the third part, it is defined the set of all incomparable elements with arbitrary but fixed c?[0,1] according to ?_T, denoted by ?I_T?^((c) ), and this set is determined for some particular t-norms. In the last part, it is constructed a family (T_? )_(??(0,1) ) of t-norms on [0,1] with the help of any t-norm T on [0,1]. Also, if T is a divisible t-norm, then it is shown that ([0,1],?_(T_? ) ) is a complete lattice. So, it is shown that there exists a t-norm T_? which is different from T_W such that (L,?_(T_? ) ) is a lattice for L=[0,1] whence answer the open problem in [35]. Also, it is shown that if T non-divisible t-norm, then (L,?_(T_? ) ) need not be a lattice.