Lesson 4

Not only products but also sums of orders can be defined. Assume the sets \(X\) and \(Y\) do not intersect; if they do, replace \(Y\) with its copy. Then, an order \(X+Y=(X \sqcup Y, \preceq)\) can be defined as follows: \(z_{1} \preceq z_{2}\) if one of the following holds:

• \(z_{1}, z_{2} \in X\) and \(z_{1} \preceq_{X} z_{2}\);


• \(z_{1} \in X\) and \(z_{2} \in Y\);


• \(z_{1}, z_{2} \in Y\) and \(z_{1} \preceq_{Y} z_{2}\).

For example, the order \(\mathbb{Z}_{\ge 0}+\mathbb{Z}_{\ge 0}\) looks as follows: \[0\prec 1 \prec 2 \prec \dotsb \prec n \prec \dotsb \prec 0' \prec 1'\prec \dotsb \ .\]

Problem. Is the sum of linear orders a linear order?