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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
Degree Granting Institution:University of Illinois at Urbana-Champaign
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 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.
Issue Date:2004
Description:106 p.
Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2004.
Other Identifier(s):(MiAaPQ)AAI3153302
Date Available in IDEALS:2015-09-28
Date Deposited:2004

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