The Property Of Power-intersected Semilattice

Lattice is widely used in topology,logic,combinatorics and algebra,and other mathematical fields.Lattice and group,ring and field are all important algebraic systems in algebra.Unlike groups,rings and fields,lattices are special partial ordered sets,As a partially ordered set.lattice can more accurately describe the incomparable things in life.A power-intersected semilattice is a intersected semilattice defined the intersection operation in a nonempty power set of lattices.It is a successful example to study the properties of power-intersected semilattice by associating it with power set.Power lattice is a hot topic in recent years,As an important part of power lattice,power-intersected semilattice is widely used in the fields of information technology,artificial intelligence and computer,The power-intersected semilattice is an algebraic structure which conforms to certain axioms and provides algebraic semantics for structural logic.Although both power-intersected semilattice and power lattice belong to lattice,their properties are different.Based on the importance of power lattice,it is necessary to study power-intersected semilattice in depth.On the basis of previous studies,this paper makes a further study on the properties of power-intersected semilattice.The article is divided into three parts:The first part: preparatory knowledge.The research significance,research status,research background and innovation of power-intersected semilattice is introduced;Some basic knowledge used in this paper is given,including:lattice,intersection semilattice,intersection homomorphism,intersection congruence,equivalence relation,power-intersected semigroups and power-intersected semilattice and so on.The second part: the property of power-intersected semilattice.The equivalence of the partial order power-intersected semilattice and algebraic power-intersected semilattice is illustrated.The necessary and sufficient conditions for the power set to be a power-intersected semilattice are proved.the direct product of power-intersected semilattices and the quotient set of the power-intersected semilattices is power-intersected semilattices are deduced.The important relationship between distribution power lattice and power semilattice is explained.According to the duality principle,the corresponding properties of the power parallel semilattice are obtained.The third part:intersection homomorphism and intersection congruence of power-intersected semilattice.It is proved out that union set of any number of intersection congruences are intersection congruences.Some concrete intersection congruences on power-intersected semilattice is gived.It is pointed out that the class of intersection congruence of power-intersected semilattice is subintersection semilattice.It is introduced that the intersection congruence induced by?has union preserving and order preserving on power-intersected semilattice.Through the intersection homomorphism of the intersection semilattice,the concrete expression of the intersection congruence of power-intersected semilattice.is constructed.According to the duality principle,the related properties of thesimultaneous homomorphism and the simultaneous congruence of the power parallel semilattices are given.