I have been studying Freyd's algebraic theory of the reals in the past week. I have problems understanding his linear representation theorem (Theorem 8.1) that every scale can be embedded in a product of linear scales. To wit, I have two problems:
I thought that scale was linearly ordered under the relation of partial order defined by $a \multimap b = \top$ (top here should be the \top of tautology). So the embedding should be trivial.
In the proof, he shows that the meet $y \vee x$ is strictly smaller than $\top$ in a scale that is a SDI (an algebra is a SDI "if whenever it is embedded into a product of algebras one of the coordinate maps is itself an embedding"). This I understand. But he also says that from this follows the disjonction property $x \multimap y \vee y \multimap x = \top$ in a SDI scale. Now, I can't see how this follows.
Thank you for your help,
Jeff.