Sums of finitely many ordinals of various kinds (Q1095143)

The addition of ordinal numbers is in general non-commutative. The paper deals with the question how many different results a sum of ordinals can have if the summands are permuted. \(E_ n\) (respectively \(L_ n\), \(S_ n\), \(T_ n\) or \(P_ n)\) is the set of all natural numbers k, for which there exist n nonzero ordinals (respectively, limit ordinals, successor ordinals, infinite successor ordinals, or pairwise-noncommutative ordinals) such that their sum assumes exactly k values if the order of the summation is permuted. The main result is that for all \(n\geq 1\), \(E_ n=L_ n=S_ n=T_ n\) and min \(P_ n=n\) and max \(P_ n=\max E_ n\). Some of these magnitudes are computed explicitly for small n.