Abstract
In a context of lattice-valued functions (also called lattice-valued fuzzy sets), where the codomain is a complete lattice L, an equivalence relation defined on L by the equality of related cuts is investigated. It is known that this relation is a complete congruence on the join-semilattice reduct of L. In terms of residuated maps, necessary and sufficient conditions under which this equivalence is a complete congruence on L are given. In the same framework of residuated maps, some known representation theorems for lattices and also for lattice-valued fuzzy sets are formulated in a new way. As a particular application of the obtained results, a representation theorem of finite lattices by meet-irreducible elements is given.
Highlights
Using a residuated mapping f from L to the power set of X ordered dually to inclusion, we identify a wide class of complete lattices for which the kernel of f is a complete congruence on L
We prove that for every complete congruence ∼ on a complete lattice L there is a nonempty subset M of L and a map μ : L → (M), so that ∼ is the kernel of a residuated map determined by the cuts of μ
Starting from a complete lattice L, and using the framework of residuated maps induced by L-valued functions μ : X → L we analyze complete congruences on L
Summary
The present investigation deals with ordered structures, mostly lattices and related functions. One of the basic tools for investigation of structure properties are p-cuts (cuts) and this approach is correctly implemented only with the complete lattices and ordering relation connected with the lattice operations. The second tool that we use in this investigation are residuated maps, closely related to Galois connections (in its definition that includes monotonicity). They are used in order theory, having a similar role as homomorphisms in the field of algebraic structures. Cabrera et al investigate Galois connections in the framework of fuzzy-preordered structures using particular fuzzy equivalence relations with a residuated lattice as the membership-values structure [15,16,17]
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