Interlacing addition of ordinals Given ordinals $\alpha,\beta$, one definition of $\alpha+\beta$ is as the order type of the disjoint union $\alpha\sqcup\beta$ ordered with all the elements of $\alpha$ before the elements of $\beta$. But this looks like only one point on a spectrum of possible orders, where maybe you interlace $\alpha$ and $\beta$ in the order. This motivates the following definition:

Let $S:=S_{\alpha,\beta}$ be the collection of order types of well orders $\prec$ of $\alpha\sqcup\beta$ satisfying $x\prec y\leftrightarrow x<y$ for all $x,y\in\alpha$ and for all $x,y\in\beta$.

I would like to characterize this set. Some observations and partial results:


*

*Obviously $\alpha+\beta\in S$ and $\beta+\alpha\in S$, and furthermore $\alpha_1+\beta+\alpha_2\in S$ when $\alpha=\alpha_1+\alpha_2$, and similarly for other finite additive decompositions of $\alpha$ and $\beta$.

*What upper and lower bounds can we find on elements of $S$? If $\sigma\in S$, then $\max(\alpha,\beta)\le\sigma$ because both $\alpha$ and $\beta$ order-inject into $\sigma$. There is of course also the trivial upper bound $\sigma<|\alpha+\beta|^+$.

*The finite case $\alpha=m$, $\beta=n$ is trivial - $S=\{m+n\}$, but if $\alpha=n$ and $\beta$ is infinite then $S=[\beta,\beta+n]$, because we can "hide" some elements of $\alpha$ behind the first copy of $\omega$ in $\beta$ and put the rest after. But even proving $S\subseteq [\beta,\beta+n]$ in this simple case looks to be a bit tricky.

*The cofinality of $\sigma\in S$ is determined by the subsets $A\cup B=\sigma$ (where $A$ and $B$ are the disjoint copies of $\alpha$ and $\beta$ in $\sigma$). If $\sup A<\sup B$, then $\operatorname{cf}\sigma=\operatorname{cf}\beta$. If $\sup A=\sup B$, then $\operatorname{cf}\alpha=\operatorname{cf}\beta=\operatorname{cf}\sigma$. Thus $\operatorname{cf}\sigma\in\{\operatorname{cf}\alpha,\operatorname{cf}\beta\}$ for all $\sigma\in S$.

 A: The way to deal with this problem is to look at the decompositions of $\alpha$ and $\beta$ in terms of indecomposables (i.e., powers of $\omega$), that is, we consider the Cantor normal form of both ordinals.
For example, consider $\omega^3+\omega+2$ and $\omega^2+5$. The only well-orderings you can form from them are $\omega^3+\omega+n$ for $n=2,\dots,7$, $\omega^3+\omega^2+\omega+n$ for $n=2,\dots,7$, and $\omega^3+\omega^2+m$ for $m=5,6,7$.
The point is that an ordinal $\gamma$ being a power of $\omega$ is equivalent to the fact that if $\gamma=\delta+\epsilon$, then $\epsilon=0$ or $\gamma$. In the well-orderings you are considering, you are splitting each indecomposable $\gamma$ mentioned in the Cantor normal forms of $\alpha$ and $\beta$ as a sum, and interleaving the pieces. 
This problem has been considered before. The key reference is 

MR0065907 (16,502a). Toulmin, G. H. Shuffling ordinals and transfinite dimension. Proc. London Math. Soc. (3) 4, (1954). 177–195. 

Toulmin says that an ordinal $\gamma$ is a shuffle of two ordinals $\alpha$ and $\beta$ if and only if $\gamma$ can be written as a disjoint union $\gamma=A\cup B$ where $A$ has order type $\alpha$ and $B$ has order type $\beta$. This is precisely what you call an interlacing sum.
Among others, he proves that the number of ordinals that are shuffles of $\alpha$ and $\beta$ is finite (see Theorem 1.38 in Toulmin's paper, that completely describes all ordinals in $S_{\alpha,\beta}$). 
The largest shuffle is the Hessenberg natural sum (which is commutative), where the indecomposable terms in the Cantor sums of $\alpha$ and $\beta$ are organized in decreasing order: Say that 
$$\alpha=\omega^{a_1}n_1+\dots+\omega^{a_k}n_k$$ and $$\beta=\omega^{a_1}m_1+\dots+\omega^{a_k}m_k,$$
where $a_1>a_2>\dots>a_k$ are ordinals and the $n_i$ and $m_i$ are natural numbers (possibly zero). Using this notation, the Hessenberg sum of $\alpha$ and $\beta$ is $$\alpha\#\beta=\omega^{a_1}(n_1+m_1)+\dots+\omega^{a_k}(n_k+m_k).$$
The smallest shuffle can be described as follows: Let $\alpha=\alpha_1+\alpha_2$ where $\alpha_1$ is zero or a limit ordinal and $\alpha_2$ is finite, and define $\beta=\beta_1+\beta_2$ analogously. Then the smallest shuffle of $\alpha$ and $\beta$ (the lower sum, in Toulmin's terminology), is 


*

*$\alpha$, if $\alpha_1>\beta_1$,

*$\alpha+\beta_2=\beta+\alpha_2$, if $\alpha_1=\beta_1$, and

*$\beta$, if $\alpha_1<\beta_1$.


There is an earlier reference dealing with a related, but different problem:

MR0005878 (3,225a). Carruth, Philip W. Arithmetic of ordinals with applications to the theory of ordered Abelian groups. Bull. Amer. Math. Soc. 48, (1942), 262–271.

Carruth focuses on what he calls natural sums (Lipparini refers to them as "mixed sums", to avoid confusion with the Hessenberg sum, that is usually also called the natural sum). He describes natural sums axiomatically: A binary operation $\oplus$ on ordinals is a natural sum if and only if it satisfies the following properties:


*

*$\oplus$ is commutative and associative.

*For any $\alpha$, $\alpha\oplus 0=\alpha$.

*For any ordinals $\alpha$, $\beta$, and $\gamma$, we have that $\alpha<\beta$ if and only if $\alpha\oplus\gamma<\beta\oplus\gamma$.


He proceeds to prove that the smallest natural sum is the Hessenberg sum, and along the way also shows that the Hessenberg sum is the largest shuffle.

(Actually, it was useful to see this question, since it helped me sort out what Carruth's paper does. I learned of it through another paper which suggested Carruth's paper only dealt with shuffles.) 
