The representability number of a chain

For each pair of linear orderings (L,M), the representability number rep_M(L) of L in M is the least ordinal \alpha such that L can be order-embedded into the lexicographic power M^\alpha. The case M = R is relevant to utility theory. The main results in this paper are as follows. (i) If \kappa is a regular cardinal that is not order-embeddable in M, then \rep_M(\kappa) = \kappa; as a consequence, \rep_\R(\kappa) = \kappa for each \kappa less or equal then \omega_1. (ii) If M is an uncountable linear ordering with the property that th lexicographic product Ax2 is not order-embeddable in M for each uncountable A \subseteq M, then \rep_M(M_\lex^\alpha)= \alpha for any ordinal \alpha; in particular, \rep_R(R_\lex^\alpha)= \alpha. (iii) If L is either an Aronszajn line or a Souslin line, then \rep_R(L) = omega_1.