A partial

order relation on a set X (Poset) is a binary relation "[less than or equal to]" on X which satisfies conditions reflexivity, antisymmetry and transitivity.

[??] x y z x {x,y} {x,y} z y {x,z} {x,z} z z z z z and the

order relation defined as [less than or equal to]: {(x,x),(y,y),(z,z),(z,x),(z,y)}.

We denote by [xi](S) the set of all fuzzy subsets of S and define an

order relation "[subset or equal to]" on the fuzzy subsets of [xi](S) as follows:

Correlation order as an important stochastic

order relation was first introduced by Joe [1]; Dhaene and Goovaerts [2] studied the bivariate case with the same marginals.

In the above studies, either the partial

order relation on policy sets is required to satisfy, or every total order subset in policy sets must have an upper bound or certain convexity condition.

Unless otherwise mentioned, throughout this paper we let E denote a partially ordered real normed linear space with an

order relation [??] and the norm [parallel]*[parallel] in which the addition and the scalar multiplication by positive real numbers are preserved by [??].

Decision-Making Based on

Order Relation. Upon obtaining attribute weight, comprehensive decision-making information is assembled after nondimensionalisation from attribute dimensions, per the [NDNWAA.sup.w] operator in Formula (3), forming a decision-making information matrix [mathematical expression not reproducible] constituted by a single solution dimension.

Order relation between two intervals [??] and [??] can be explained in two ways; first one is an extension of < on real line, that is, [mathematical expression not reproducible] iff [a.sup.R] < [b.sup.L], and the other is an extension of the concept of set inclusion, that is, [mathematical expression not reproducible] and [a.sup.R] < [b.sup.R].

Most of the surveyed algebras have essentially an

order relation (i.e.

Similarly, (Wei Zhou et al., 2013) proposed

Order Relation Vector Model for calculating QoS preference based on four QoS attributes such as Price, Response time (the delay), Reliability and Reputation.

Remember that for the desired reductio to work, whatever is necessary for Lewis' theory of intra-world causation must be sufficient for trans-world causation; and while Lewis' theory does not require anything stronger than a weak

order relation of closeness among worlds, this is not sufficient to construct the kind of closeness relation among pairs of worlds necessary for trans-world causation.