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Title:  Lexicographic Products of Linear Orderings 
Author(s):  Giarlotta, Alfio 
Doctoral Committee Chair(s):  Henson, C. Ward; Stephen Watson 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Economics, Theory 
Abstract:  For each pair of linear orderings (L, M), the representability number reprM(L) of L in M is the least ordinal alpha such that L can be orderembedded into the lexicographic power Malex . The case M = R is relevant to utility theory, a branch of mathematical economics. First we characterize lexicographic products whose representability number in R is 1. Next we prove the following results: (i) if kappa is a regular cardinal which is not orderembeddable in M, then reprM(kappa) = kappa; as a consequence, reprR (kappa) = kappa for each kappa ≥ o1; (ii) if M is an uncountable linear ordering with the property that A xlex 2 is not orderembeddable in M for each uncountable A ⊆ M, then repr M( Malex ) = alpha for any ordinal alpha; in particular, reprR ( Ralex ) = alpha; (iii) if L is either an Aronszajn line or a Souslin line, then reprR (L) = o1. We also study representations of linear orderings by means of trees. We prove the following fact: if alpha is an indecomposable ordinal and L is a linear ordering such that neither alpha nor its reverse ordering alpha* orderembed into L, then L embeds into the lexicographic linearization of a binary tree having no branch of length alpha. Finally we study the class of small chains, i.e., the linear orderings that orderembed neither o 1 nor o1* nor an Aronszajn line. We construct a sequence of small chains with increasing lexicographic complexity and with representability number in R as large as o1. 
Issue Date:  2004 
Type:  Text 
Language:  English 
Description:  106 p. Thesis (Ph.D.)University of Illinois at UrbanaChampaign, 2004. 
URI:  http://hdl.handle.net/2142/86835 
Other Identifier(s):  (MiAaPQ)AAI3153302 
Date Available in IDEALS:  20150928 
Date Deposited:  2004 
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Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois