The suprume and infium of a family of proper ideals and their properties

The suprume and infium of a family of proper ideals on anonempty set are studied.Meanwhile,the method of monad theory is used in the nonstandardenlarged model.And,the definition of the suprume and infium of a family ofproper ideals are introduced and some necessary and sufficient conditions oftheir existence are discussed.Furthermore,the properties of suprumeand infium of a family of proper ideals are obtained.