完备格上预余拓扑的确定

通过定义完备格L上的预余拓扑、预闭包算予、预外导算子族、预内部算子、覆盖族及它们上的序关系，研究给定完备格L上预余拓扑的确定方式．证明了（1）给定完备格L上预余拓扑的全体、预闭包算子的全体构成了彼此同构的完备格；（2）当L是闭集格时，L上预外导算子族的全体与预余拓扑的全体构成了彼此同构的完备格；（3）当L是完备DeMorgan代数时，L上的预余拓扑的全体、预内部算子的全体、覆盖族的全体构成了彼此同构的完备格．因此给定完备格L上的预余拓扑可以由L上的预闭包算子、预外导算子族、预内部算子及覆盖族确定．

Determination of pre-cotopologies on a complete lattice

The notions of pre-cotopology, pre-closure operator, family of pre-outside-derived operators, pre-interior operator, family of covers, and their orders on a given complete lattice are introduced, and the way to determine pre-cotopologies on a complete lattice is studied. It is proved that, for a given com- plete lattice L, （1） Both （CT（L） ,C） （the set of all pre-cotopologies on L） and （CL（L）, ≤） （the set of all pre-closure operators on L） are complete lattices which are isomorphic each other. （2） If L is a closed- set lattice, then both （CT（L） ,C） and （FD（L）, ≤） （the set of all families of pre-outside-derived operators on L） are complete lattices which are isomorphic each other. （3） If L is a complete DeMorgan algebra, then both （IN（L）, ≤） （the set of all pre-interior operators） and （FC（L）, ≤） （the set of all families of covers on L） are complete lattices which are isomorphic with （CT（L）,C）. Thus a pre-cotopology on a given complete lattice may be determininated by a pre-closure operator, a family of pre-outside-derived operators, a pre-interior operator, or a family of covers.