## Files in this item

FilesDescriptionFormat

application/pdf

3153302.pdf (7MB)
(no description provided)PDF

## Description

 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 Urbana-Champaign 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 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 Type: Text Language: English Description: 106 p.Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2004. URI: http://hdl.handle.net/2142/86835 Other Identifier(s): (MiAaPQ)AAI3153302 Date Available in IDEALS: 2015-09-28 Date Deposited: 2004
﻿