The representability number of a chain

For each pair of linear orderings (L,M), the representability number reprM(L) of L in M is the least ordinal α such that L can be order-embedded into the lexicographic power Mlexα. 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 reprM(κ)=κ; as a consequence, reprR(κ)=κ for each κ⩾ω1. (ii) If M is an uncountable linear ordering with the property that A×lex2 is not order-embeddable in M for each uncountable A⊆M, then reprM(Mlexα)=α for any ordinal α; in particular, reprR(Rlexα)=α. (iii) If L is either an Aronszajn line or a Souslin line, then reprR(L)=ω1.