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https://hdl.handle.net/2142/86835
Description
Title
Lexicographic Products of Linear Orderings
Author(s)
Giarlotta, Alfio
Issue Date
2004
Doctoral Committee Chair(s)
Henson, C. Ward
Stephen Watson
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Economics, Theory
Language
eng
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 order-embedded 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 order-embeddable 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 order-embeddable 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* order-embed 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 order-embed 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.
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