The Properties Of The Order Dual And Quotient Space Of Lattice Algebra

In the traffic information engineering and control, traffic information security is an important research field in which confidentiality and authentication are two basic attributes. The lattice-based public key cryptosystem and digital signature scheme with good safety and low bandwidth have received great attention in recent years. In this paper our main research object is lattice algebra which is a vector space with ordered and algebraic structure. The special lattice algebra can be used to design cryptographic algorithms or digital signature scheme. The research has important theoretical significance and potential applications.This paper is devoted to the theoretical study on order dual of a lattice algebra, the f-module and d-module on a lattice algebra and the quotient of lattice algebra. Actually, the present work consists of four main parts.In the first part, we investigate the ordered properties of multiplicative spaces of order dual of the lattice algebra, especially the f-algebra, the almost f-algebra and the d-algebra. It’s proved that under some conditions these multiplicative spaces equipped with Arens multiplications are order ideals in the order dual of lattice algebra. And as an application, a necessary and sufficient condition is derived under which the continuous order bidual of an f-algebra to be semiprime.In the second part, the orthomorphisms, the f-orthomorphisms and the f-linear operators on the order dual of an f-algebra are considered. We introduce the new concepts of f-orthomorphisms and the f-linear operators. Then we discuss the ordered properties of these operators’spaces, and study the relationships between these operators. It is proved that the orthomorphisms must be f-linear operators, and f-linear operators must be f-orthomorphisms. Especially, when the f-algebra is square root closed, they are the same class of operators. As a result, we give a new characterization for the orthomorphisms.In the third part, the f-module and d-module on a lattice algebra have been studied. Firstly, the relationship between the disjointness preserving operators and linear operators on an f-module is discussed. It is obtained that these operators are equvalent under some conditions. Then the linear proerty of the inverse of a disjointness preserving operator is given. Moreover we introduce the concept of the d-module, and obtain that the order bidual of a d-module is likewise a d-module. We study the d-modules over the ideals generated by a lattice homomorphism or an interval preserving operator with some examples.Finally, the quotient space of a lattice algebra and their algebraic properties have been fully studied. The concepts of the quotient lattice algebra, especially the quotient f-algebra, quotient almost f-algebra and quotient d-algebra are given, as well as the equivalent characterizations. We also investigate the algebraic properties of a quotient lattice algebra. The examples are given to compare these properties. The equivalent condition under which a quotient lattice algebra is semiprime is obtained. At last we discuss the commutative property and show the conditions under which the unit and the inverse in a quotient f-algebra exist.