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In [[algebra]], an ''ordered semigroup'' is a [[semigroup]] (''S'',•) together with a [[partial order]] ≤ that is ''compatible'' with the semigroup operation, meaning that ''x'' ≤ ''y'' implies z•x ≤ z•y and x•z ≤ y•z for all ''x'', ''y'', ''z'' in ''S''. If S is a [[Group_(mathematics)|group]] and it is ordered as a semigroup, one obtains the notion of [[ordered group]], and similarly if S is a [[monoid]] it may be called ''ordered monoid''. [[Ordered vector space|Partially ordered vector space]]s and [[Riesz space|vector lattice]]s are important in [[multiobjective optimization|optimization with multiple objectives]].
== Ordinal optimization in
{{See also|Selection algorithm}}
Problems of ordinal optimization arise in many disciplines. [[Computer science|Computer scientists]] study [[selection algorithm]]s, which are simpler than [[sorting algorithm]]s.<ref>[[Donald Knuth]]. ''[[The Art of Computer Programming]]'', Volume 3: ''Sorting and Searching'', Third Edition. Addison-Wesley, 1997. ISBN 0-201-89685-0. Section 5.3.3: Minimum-Comparison Selection, pp.207–219.</ref><ref>[[Thomas H. Cormen]], [[Charles E. Leiserson]], [[Ronald L. Rivest]], and [[Clifford Stein]]. ''[[Introduction to Algorithms]]'', Second Edition. MIT Press and McGraw-Hill, 2001. ISBN 0-262-03293-7. Chapter 9: Medians and Order Statistics, pp.183–196. Section 14.1: Dynamic order statistics, pp.302–308.</ref>
[[Statistical decision theory]] studies "selection problems" that require the identification of a "best" subpopulation or of identifying a "near best" subpopulation.<ref>Gibbons, Jean Dickinson; [[Ingram Olkin|Olkin, Ingram]], and Sobel, Milton, ''Selecting and Ordering of Populations'', Wiley, (1977). (Republished as a Classic in Applied Mathematics by SIAM.)</ref><ref>{{cite book|last1=Gupta|first1=Shanti S.|last2=Panchapakesan|first2=S.|title=Multiple decision procedures: Theory and methodology of selecting and ranking populations|series=Wiley Series in Probability and Mathematical Statistics|publisher=John Wiley & Sons|___location=New York|year=1979|pages=xxv+573|isbn=0-471-05177-2|id={{MR|555416}}|}} (Republished as a Classic in Applied Mathematics by SIAM.)</ref><ref>Santner, Thomas J., and Tamhane, A. C., ''Design of Experiments: Ranking and Selection'', M. Dekker, (1984). </ref><ref>Robert E. Bechhofer, Thomas J. Santner, David M. Goldsman. ''Design and Analysis of Experiments for Statistical Selection, Screening, and Multiple Comparisons''. John Wiley & Sons, 1995.</ref><ref>Friedrich Liese, Klaus-J. Miescke. 2008. ''Statistical Decision Theory: Estimation, Testing, and Selection''. Springer Verlag.</ref>
== Applications ==
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