, Volume 22, Issue 2, pp 145183
First online:
Regulating Functions on Partially Ordered Sets
 Venkat AnantharamAffiliated withDepartment of Electrical Engineering and Computer Sciences, University of California
 , Takis KonstantopoulosAffiliated withSchool of Mathematical and Computer Sciences, Heriot–Watt University Email author
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We study the socalled Skorokhod reflection problem (SRP) posed for realvalued functions defined on a partially ordered set (poset), when there are two boundaries, considered also to be functions of the poset. The problem is to constrain the function between the boundaries by adding and subtracting nonnegative nondecreasing (NN) functions in the most efficient way. We show existence and uniqueness of its solution by using only order theoretic arguments. The solution is also shown to obey a fixed point equation. When the underlying poset is a σalgebra of subsets of a set, our results yield a generalization of the classical Jordan–Hahn decomposition of a signed measure. We also study the problem on a poset that has the structure of a tree, where we identify additional structural properties of the solution, and on discrete posets, where we show that the fixed point equation uniquely characterizes the solution. Further interesting posets we consider are the poset of real nvectors ordered by majorization, and the poset of n × n positive semidefinite real matrices ordered by pointwise ordering of the associated quadratic forms. We say a function on a poset is of bounded variation if it can be written as the difference of two NN functions. The solution to the SRP when the upper and lower boundaries are the identically zero function corresponds to the most efficient or minimal such representation of a function of bounded variation. Minimal representations for several important functions of bounded variation on several of the posets mentioned above are determined in this paper.
Key Words
Skorokhod reflection partially ordered sets monotonicity Jordan decomposition signed measures majorization positive semidefinite matrices bounded variationMathematics Subject Classifications (2000)
Primary 06A06, 06B35, 26A45, 47H09, 47H10 Secondary 06B23, 15A57, 60K25, 60J65 Title
 Regulating Functions on Partially Ordered Sets
 Journal

Order
Volume 22, Issue 2 , pp 145183
 Cover Date
 200505
 DOI
 10.1007/s1108300590149
 Print ISSN
 01678094
 Online ISSN
 15729273
 Publisher
 Springer Netherlands
 Additional Links
 Topics
 Keywords

 Skorokhod reflection
 partially ordered sets
 monotonicity
 Jordan decomposition
 signed measures
 majorization
 positive semidefinite matrices
 bounded variation
 Primary 06A06, 06B35, 26A45, 47H09, 47H10
 Secondary 06B23, 15A57, 60K25, 60J65
 Authors

 Venkat Anantharam ^{(1)}
 Takis Konstantopoulos ^{(2)}
 Author Affiliations

 1. Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA, 94720, USA
 2. School of Mathematical and Computer Sciences, Heriot–Watt University, Edinburgh, EH14 4AS, UK