Because addition and multiplication of two order types are non-commutative operations, we have that for every integer n, given n ordinals, there are at most n! distinct possible values for the sum (or the product) of these n ordinals when all permutations are considered.
In fact, A. Wakulic (Fund. Math., XXXVI, pp 255-260) was able to establish the effective function F(n) giving the maximal possible number of distinct values for the SUM of n ordinals, when considering all permutations.
QUESTION: What is the effective function G(n) giving the maximal possible number of distinct values for the PRODUCT of n ordinals, when considering all permutations ?
Gérard Lang 



What is the maximal number of distinct values of the product of n permuted ordinals
Simply take the ordinals $\omega+1,...,\omega+n$ and one obtains $n!$ distinct products (This solution was taken from Chapter 8 Problem 39 and Chapter 9 Problem 66 in the book Problems and Theorems is Classical Set Theory).


This follows from the easily provable fact that for natural numbers $r\_{1},...,r\_{n}$ we have $$(\omega+r\_{1})\cdots(\omega+r\_{n})=\omega^{n}+\omega^{n-1}r\_{n}+\omega^{n-2}r\_{n-1}+...+\omega r\_{2}+r\_{1}.$$


