On λ-invariants associated to linear extensions of the product ordering on Nd

We introduce the λ-invariant associated to a linear extension of the product ordering on Nd, then show that it satisfies properties analogous to those obtained by Satoh in [9] and Rosenberg in [8]. We also show how our results can be used to recover the results proved by Barman and Saikia in [2] and [1], and to prove an analogue of the result bounding the λ-invariant of a pseudo-polynomial obtained in [8] for each of the λ-invariants discussed in [1].