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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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