Abstract
A hierarchy of chains is a transfinite sequence of linear orderings such that each chain in the sequence order-embeds into all chains following it but not in those preceding it. We construct a c + -long hierarchy of chains that order-embed into the lexicographic power \(({\mathbb{R}}^\omega,\prec_{\rm lex})\). Each linear ordering L in this hierarchy is such that there exists a tree representation of L, which is an ℝ-branching tree with no infinite branches. The existence of such a hierarchy sheds some light on the hidden complexity of \(({\mathbb{R}}^\omega,\prec_{\rm lex})\).
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References
Caserta, A., Giarlotta, A., Watson, S.: Debreu-like properties of utility representations. J. Math. Econ. 44, 1161–1179 (2008)
Chipman, J.P.: On the lexicographic representations of preference orderings. In: Chipman, J.S., Hurwicz, L., Richter, M., Sonnenschein, H.F. (eds.) Preference, Utility and Demand, pp. 276–288. Harcourt Brace and Jovanovich, New York (1971)
Fleischer, I.: Embedding linearly ordered sets in real lexicographic products. Fundam. Math. 49, 147–150 (1961)
Giarlotta, A.: Lexicographic products of linear orderings. Ph.D. dissertation, University of Illinois at Urbana-Champaign, Urbana (2004)
Giarlotta, A.: The representability number of a chain. Topol. its Appl. 150, 157–177 (2005)
Harzheim, E.: Einbettung totalgeordnete mengen in lexicographische produkte. Math. Ann. 170, 245–252 (1967)
Herden, G., Mehta, G.B.: The Debreu Gap Lemma and some generalizations. J. Math. Econ. 40, 747–769 (2004)
Holland, W.C., Kuhlmann, S., McCleary, S.: Lexicographic exponentiation of chains. J. Symb. Log. 70, 389–409 (2005)
Knoblauch, V.: Lexicographic orders and preference representation. J. Math. Econ. 34, 255–267 (2000)
Kuhlmann, S.: Isomorphisms of lexicographic powers of the reals. Proc. Am. Math. Soc. 123/9, 2657–2662 (1995)
Kunen, K.: Set Theory. An Introduction to Independence Proofs. North-Holland, Amsterdam (1980)
Mitchell, B.: Theory of Categories. Academic Press, New York (1965)
Rosenstein, J.G.: Linear Orderings. Academic Press, New York (1982)
Todorčević, S.: Trees and linearly ordered sets. In: Kunen, K., Vaughan, J. (eds.) Handbook of Set Theoretic Topology, pp. 235–293. North-Holland, Amsterdam (1984)
Wakker, P.: Continuity of preference relations for separable topologies. Int. Econ. Rev. 29, 105–110 (1988)
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Giarlotta, A., Watson, S. A Hierarchy of Chains Embeddable into the Lexicographic Power \(\boldsymbol{({\mathbb{R}}^\omega,\prec_{\rm lex})}\) . Order 30, 463–485 (2013). https://doi.org/10.1007/s11083-012-9256-2
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DOI: https://doi.org/10.1007/s11083-012-9256-2