Fuzzy submonoids, fuzzy preorders and quasi-metrics

Let L be a residuated lattice. Then, given a monoid M, we define a Galois connection from the lattice of the compatible L-preorders in M and the lattice of L-submonoids of M. Given a set S we define a Galois connection between the lattice of the L-preorders in S and the lattice of L-submonoids of the monoid (SS,∘,i). A link with the notion of quasi-metric is also established.