Research On Some Uncertainty Problems Of Lattices And BCI-algebras

Fuzzy set theory,rough set theory and soft set theory are all useful mathematical tools to deal with uncertainty problems,and are studied by many scholars.This paper mainly considers the combination of these theories with lattices and BCI-algebras,and the relationships between fuzzy sets,rough sets,soft sets and lattices(BCI-algebras),and studies some uncertainty problems of lattices and BCI-algebras.This paper mainly introduces the following aspects:(1)A new congruence relation based on a fuzzy ideal of a distributive lattice is introduced.Moreover,rough lattices(ideals,filters)and rough fuzzy lattices(ideals,filters)based on fuzzy ideals are investigated.In particular,we consider the relationships between lattices(ideals,filters)and rough lattice(ideals,filters),fuzzy lattices(ideals,filters)and rough fuzzy lattices(ideals,filters),respectively.Some characterizations are also discussed.(2)By means of a new congruence relation induced by a fuzzy ideal of a distributive lattice,we introduce the definitions of the lower and upper approximations over a distributive lattice.Moreover,we study some special properties of rough sets which can be constructed by means of the congruence relations determined by fuzzy ideals over distributive lattices.Finally,the properties of generalized rough sets with respect to fuzzy ideals over distributive lattices are also investigated.(3)We generalize the concepts of rough lattices(ideals,filters)to the concepts of soft rough lattices(ideals,filters).Further,we study roughness over lattices with respect to soft approximation spaces.Some new soft rough operations over lattices are explored.In particular,lower and upper soft rough lattices(ideals,filters)with respect to another soft set are also investigated.(4)We generalize the concepts of soft rough lattices(ideals,filters)and rough fuzzy lattices(ideals,filters)to soft rough fuzzy lattices(ideals,filters).Moreover,we study roughness over lattices with respects to a soft approximation space.Some new soft rough fuzzy operations over lattices are explored.In particular,lower and upper soft rough fuzzy lattices(ideals,filters)over lattices with respect to another fuzzy soft set are also investigated.(5)We generalize the concepts of soft rough BCI-algebras(ideals)to the concepts of soft rough fuzzy BCI-algebras(ideals).Moreover,we study roughness over BCIalgebras with respects a Z-soft approximation space.Some new Z-soft rough fuzzy operations over BCI-algebras are explored.In particular,Z-lower and Z-upper soft rough fuzzy BCI-algebras(ideals)are also investigated.(6)We generalize the concepts of soft rough fuzzy BCI-algebras(ideals)to the concepts of soft fuzzy rough BCI-algebras(ideals).Moreover,we study roughness over BCIalgebras with respect to a pseudo soft fuzzy approximation space.Some new soft fuzzy rough operations over BCI-algebras are explored.In particular,Z-upper and Z-lower soft fuzzy rough BCI-algebras(ideals)over BCI-algebras are also investigated.