Results of Associated Implication Algebra on a Partial Ordered Set

Some sufficient and necessary conditions that implication algebra on a partial ordered set is associated implication algebra are obtained, and the relation between lattice H implication algebra and associated implication algebra is discussed. Also, the concept of filter is proposed with some basic properties being studied.

EKELAND’S VARIATIONAL PRINCIPLE AND CARISTI’S COINCIDENCE THEOREM FOR SET-VALUED MAPPINGS IN PROBABILISTIC METRIC SPACES

By using the partial ordering method,a more general type,of Ekeland’s ariational principle and a set-valued Caristi’s coincidence theorem in probabilistic metric spaces are obtained.In addition,a direct simple proof of the equivalence between these two theorems in probabilistic metric spaces is given.

From Optimal Stopping Problems over Tree Sets to Optimal Stopping Problems over Partially Ordered Sets

In this paper, we discuss relations between optimal stopping problems over tree sets and partially ordered sets, prove that there is a 1-1 correspondence between them and so every optimal strategy can be obtained in the set of optimal control variables.

Existence and Uniqueness of Fixed Point in Partially Ordered Sets and Applications to Ordinary Differential Equations

We prove some fixed point theorems in partially ordered sets, providing an extension of the Banach contractive mapping theorem. Having studied previously the nondecreasing case, we consider in this paper nonincreasing mappings as well as non monotone mappings. We also present some applications to first-order ordinary differential equations with periodic boundary conditions, proving the existence of a unique solution admitting the existence of a lower solution.

Logical Properties of Lattice Filter of Lattice Implication Algebra

Some new properties of lattice filters are presented based on the order-preserving mapping and lattice homomorphism, and two necessary and sufficient conditions for lattice filters under the chain type are given. Then, the relations between lattice filter and lattice implication algebras （LIAs）, i. e., the relations between lattice filter and LIA-filters, and the related properties are investigated. In addition, three necessary and sufficient conditions for LIA-filters are discussed. The obtained results may serve as some theoretical supports to lattice-valued logical system.

Special Relationship among Decision Making Units based on Partially Ordered Set and New Evaluation and Projection Methods

This paper provides data envelopment analysis methods based on partially ordered set theory.These methods reveal the special relationships between two decision making units from the perspective of mathematical theory and offer the classification,projection and improvement methods of decision making units.It is proved that an efficient decision making unit must be a maximal element of the related poset,and the maximal element may not be efficient.For this,we introduce the concepts of minimum envelope and efficiency envelope which further reveal the special relationship among efficient and inefficient decision making units.Compared with the previous methods,this method not only reveals theoretically the complex relationship among decision making units and the causes of the ineffectiveness,but also gives a new importance and competitiveness measurement method to each decision making unit.Finally,related algorithm and examples are given for the application of these methods to complex decision making problems.

On Relationships Between Semi- antichains andUnichain Coverings in Discrete Mathematics

The paper explores the relationships between the largest cardinality of a semi antichain and the smallest cardinality of its unichain covering on the direct product space induced by two partially ordered sets,through studying on partially ordered sets. A sufficient condition under which they are equal is obtained.

CLOSURE AXIOMS FOR POSET MATROIDS

An excellent introduction to the topic of poset matroids is due to M, Barnabei, G, Nicoletti and L. Pezzoli. In this paper, we extensively study the closure operators of poset matroids and obtain the closure axioms for poset matroids; thereby we can characterize poset matroids in terms of the closure axioms. Some corresponding properties of combinatorial schemes are also obtained.

DATA PREORDERING IN GENERALIZED PAV ALGORITHM FOR MONOTONIC REGRESSION

Monotonic regression （MR） is a least distance problem with monotonicity constraints induced by a partiaily ordered data set of observations. In our recent publication [In Ser. Nonconvex Optimization and Its Applications, Springer-Verlag, （2006） 83, pp. 25-33], the Pool-Adjazent-Violators algorithm （PAV） was generalized from completely to partially ordered data sets （posets）. The new algorithm, called CPAV, is characterized by the very low computational complexity, which is of second order in the number of observations. It treats the observations in a consecutive order, and it can follow any arbitrarily chosen topological order of the poset of observations. The CPAV algorithm produces a sufficiently accurate solution to the MR problem, but the accuracy depends on the chosen topological order. Here we prove that there exists a topological order for which the resulted CPAV solution is optimal. Furthermore, we present results of extensive numerical experiments, from which we draw conclusions about the most and the least preferable topological orders.