The order on the rationals has an orthogonal order with the same order type
The authors study the existence of linear orders orthogonal to the order of the rationals and prove the theorem that if \(\nu \geqslant {\omega ^2}\) is a countable ordinal then the order of the rationals \(\eta = \left( {{\mathbb Q}, \leqslant } \right)\) has an orthogonal linear order of order-type \(\nu \) and \(\eta \) has an orthogonal linear order of order-type \(\eta \). \(\omega \) denotes the first infinite countable ordinal, \(\mathbb{Q}\) denotes the set of rational numbers.
scientific article; zbMATH DE number 5988083