The quantic conuclei on quantales

In this paper, the main purpose is to investigate some properties of quantic conuclei. Firstly, the concept of an ideal conucleus is introduced and a characterization for a map to be an ideal conucleus is given. Secondly, the relations between quantic nuclei and quantic conuclei are studied on Girard quantale, based on which, the extensions of quantic conuclei (nuclei) to a Girard quantale and the relations between those extensions are discussed. Finally, the concrete structure of quantale without non-trivial quantic conuclei is given and it is proved that the number of subquantales of an infinite quantale is infinite.