Researches on pre-Girard quantale and its properties

Some of the properties of pre-Girard quantale and the relation between nuclei and conuclei on pre-Girard quantale are studied,the definition of the ideal conucleus is simplified,some of the properties of ideal conuclei are discussed,and the concrete constructions of ideal conuclei on right-sided and idempotent quantales are given.A sufficient and necessary condition for a pre-Girard quantale to be a Girard quantale is obtained.It is proved that every quantale can be embedding into a pre-Girard quantale,and nuclei and ideal conuclei on pre-Girard quantale are in one-to-one correspondence.