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Henson, C. Ward Stephen Watson Giarlotta, Alfio 2015-09-28T15:19:46Z 2015-09-28T15:19:46Z 10000-01-01 2004 2004 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. Made available in DSpace on 2015-09-28T15:19:46Z (GMT). No. of bitstreams: 2 license.txt: 4848 bytes, checksum: 96035ab3f5e1c23cc7138a224ce498bd (MD5) 3153302.pdf: 6739028 bytes, checksum: ec8b12bacf11f41d80f4179488dec68f (MD5) Previous issue date: 2004 Embargo set by: Seth Robbins for item 88116 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs U of I Only 106 p. Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2004. http://hdl.handle.net/2142/86835 (MiAaPQ)AAI3153302 eng Economics, Theory Lexicographic Products of Linear Orderings text Dissertation Ph.D. Mathematics Mathematics University of Illinois at Urbana-Champaign