Construction Of Uninorms On Bounded Lattices

Uninorms introduced by Yager and Rybolov are important aggregation functions,which are a generalization of triangular norms and triangular conorms.Because of their good properties,uninorms have proved to be useful both in theory and practical applications such as aggregation of information,fuzzy system modeling,neural networks and so on.One of the most important questions of uninorms is construction.There has been a lot of results of uninorms constructing,which are focused on unit interval or finite chains.Recently,the construction of uninorms defined on bounded lattices is an area of increasing interest.For one reason,compared with the order on unit interval or finite chain,total order is not a necessary conditions on bounded lattices.On bounded lattices,uninorms can be more widely used in theory and practical applications.For another reason,uninorms on bounded lattices are generalization of uninorms on unity interval.The main concern of this paper is to construct uninorms on bounded lattices.The main work of this paper can be generalized as follows:1.Giving two method of constructing uninorms on bounded totally ordered sets.2.Introducing the concept of 1-dimensional unity interval.Giving the proof that 1-dimensional-unity-interval-mapping is an order isomorphism.Then construction a lattice isomorphism between the bounded totally ordered set which satisfied the condition of 1-dimensional-unity-interval-mapping and unity interval such that the property of uninorms on unity interval can be maintained on the bounded total ordered sets.3.Discussing the construction of uninorms on normal bounded lattices to illuminate that it is necessary to give some additional conditions when constructing the uninorms,or it is difficult to get another result.4.Giving two methods of constructing uninorms on bounded lattices sets which satisfied Ie=(?).Finding a way to turn a normal bounded lattices to a bounded lattices with Ie=(?).5.Introducing the concept of 2-dimensional unity interval.Giving the proof that 2-dimensional-unity-interval-mapping is an order isomorphism.Then construction a lattice isomorphism between the bounded lattices which satisfied the condition of 2-dimensional-unity-interval-mapping and unity interval such that the property of uninorms on unity interval can be maintained on the bounded total ordered sets.