Nuclei and conuclei on residuated lattices

In this paper, we investigate some properties of nuclei and conuclei on residuated lattices and their concrete structures. By means of nuclei and conuclei we give the respective characterizations for a residuated lattice reducing to a generalized MV-algebra and a generalized BL-algebra. Based on studying regular commutative bounded integral residuated lattices, we introduce the concept of a pseudo-dual quantale and investigate some properties of it, and prove that the set of fuzzy sets of a nonempty set equipped with multiplication and negation operations is a commutative Girard quantale. Finally, we discuss the relationship between quantic nuclei and quantic conuclei on a pseudo-dual quantale, and show that the quantic nuclei and the ideal conuclei on a pseudo-Girard quantale are in one-to-one correspondence.