Studies Of Relative Questions About Linear Projective Spaces Of Linear FS-lattices

Domain theory appeared in the early part of 1970s when D. Scottwas led by problems of semantics for computer languages to initiated the studyof continuous lattices. The main objects in domain theory are partially orderedsets(posets for short)and the mappings between posets. Function spaces areone of the most important objects in Domain theory, then projective spaces areimportant function spaces. We investigate the properties of linear projectivespaces on linear FS-lattice,linear projection lattice. Moreover we study therelation of linear projection lattice. We obtain several important results:We prove that linear projective space of linear FS-lattice is an algebraiclattice if and only if for any map in linear projective space of linear FS-lattice,its image is an algebraic lattice if and only if any map in linear projective spaceof linear FS-lattice, its image is an algebraic linear FS-lattice. Linear projectivespace of completely distributive lattice is a continuous lattice if and only if itis a strongly algebraic lattice such that its strongly compact set has no orderdense chain if and only if its linear projective space is an powerset lattice of oneset, further clear the construction of linear projective space. Linear FS-latticeare closed under linear projective space and Cartician product. We define linearprojection lattice, we illustrate the relation linear projection lattice and lineardomain. The category of linear FS-lattice is equivalent to a category of projectiondomain. Especially, any linear projection lattice is isomorphic to Scott closedlattice of some projection domain; when projection domain is a complete lattice,its projective space is a continuous retract of linear projection lattice of Scott closed lattice.