A T-partial order obtained from t-norms

The main aim of the present thesis is to determine the conditions on t-norm T for the subset H_T consisting of all idempotent elements of t-norm T on a complete lattice L and to investigate some conditions for the set A of all elements of a complete lattice L which are in the form of the supremum of any family of atoms to be a complete lattice with respect to the order obtained from t-norms.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 three parts. In the first part, firstly, the notion T-partial order, denoted by ?_T, on a bounded lattice L by means of a t-norm on L is given. Also, in this part, some connections between the orders ?_T and ? are investigated. In the second part, it is shown that L need not be a chain (or lattice) with respect to ?_T even if L is a chain (or lattice). Also, it is determined a necessary condition making L a lattice with respect to ?_T. In the last part, some conditions are determined on t-norms for the set H_T of all idempotent elements of t-norm T to be a complete lattice with respect to the order ?_T. Also, it is proved that for an integral, commutative, residuated l- monoid M=(L,?,? ), if M is divisible, then the subset H_T of all idempotent elements with respect to ? forms a Heyting algebra, and the implication in H_T coincides with the implication based on ?. So, it is obtained from this conclusion that the algebraic strong De Morgan?s law is not necessary for the proof of the Teorem in the study of Drossos. Also, it is examined that some conditions making the subset A of all elements of L which are in the form of the suremum of any family of atoms a complete lattice.Key Words: T-Norm, Bounded lattice, ?- distributive, Idempotent element, Atom