The representability number of a chain

For each pair of linear orderings ( L , M ) , the representability number repr M ( L ) of L in M is the least ordinal α such that L can be order-embedded into the lexicographic power M lex α . The case M = R is relevant to utility theory. The main results in this paper are as follows. (i) If κ is a regular cardinal that is not order-embeddable in M, then repr M ( κ ) = κ ; as a consequence, repr R ( κ ) = κ for each κ ⩾ ω 1 . (ii) If M is an uncountable linear ordering with the property that A × lex 2 is not order-embeddable in M for each uncountable A ⊆ M , then repr M ( M lex α ) = α for any ordinal α; in particular, repr R ( R lex α ) = α . (iii) If L is either an Aronszajn line or a Souslin line, then repr R ( L ) = ω 1 .