T-norms obtained via the irreducible elements in lattices

In this thesis, the structure of the poset of -irreducible ( -irreducible) elements of the direct product of complete lattices is observed. Particularly, for a given t-norm (t-conorm) on the complete lattice , the conditions under which the restriction of the t-norm to the set of -irreducible ( -irreducible) elements is again a t-norm (t-conorm) are researched. A method for constructing t-norms (t-conorms) on finite distributive lattice from a given behaviour on the -irreducible ( -irreducible) elements is presented. It is proved that a t-norm on a distributive lattice is idempotent if and only if it is idempotent on the set of the -irreducible elements. A method to construct t-conorms on a dual ideal of a complete lattice via given t-conorms on the lattice is given. A partition of the set of -irreducible elements of the lattice L^[n] is given and a method to construct t-norms on the lattice is presented using this partition. Also some special elements in a lattice defined by the operations and are generalized by t-norms (t-conorms). Some algebraic properties of these elements are investigated. A characterization of the automorphisms of the direct product of bounded chains is given. New t-norms are constructed from given t-norms by automorphisms.