| A lattice is an important algebra with wide applications in many areas such as algebras, geometries, computer sciences, economics, managements and so on. Fuzzy sets, soft sets and fuzzy soft sets are useful tools for studying algebra structures. In this paper, the theories of soft sets and fuzzy soft sets are applied to lattice algebras. Fuzzy soft lattices (ideals) are introduced and studied. This paper is organized as follows:In the first part, we study fuzzy soft lattices and fuzzy soft ideals in lattices. Firstly, using level sets of fuzzy soft sets, we introduce the notions of fuzzy soft lattices and fuzzy soft ideals and give some equivalent conditions of them. Next, using existing or new operations of fuzzy soft sets, we discuss some operation properties of fuzzy soft lattices (ideals). Through research, we find that the fuzzy soft lattices (ideals) are closed under "join","meet" and "and" operations. Finally, using the extension principle of fuzzy sets and the fuzzy soft homomorphism of lattices, we study some properties of the images and inverse images of fuzzy soft lattices (ideals). Under the fuzzy soft homomorphism (φ,φ), the image and inverse image of fuzzy soft lattices are still fuzzy soft lattices. In particular, if (φ,φ) is a surjective, the image and inverse image of fuzzy soft ideals are still fuzzy soft ideals.In the second part, we propose generalized fuzzy soft structures of lattices, that is (∈,∈∨q)-fuzzy soft lattices and (λ,μ)-fuzzy soft lattices. And we present several characteriza-tions of them by using corresponding generalized fuzzy substructures. In the same way, generalized fuzzy soft ideals are introduced and some basic properties of them are investigated.Fuzzy soft lattices (ideals) introduced by us are not only the generalizations of fuzzy sublattices (ideals) in [33-35], but also that of soft lattices (ideals) in [32] and classical lattices theory in [29]. Some results obtained further develop related theories of fuzzy algebras. |
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