I have a question about notation in a book I'm reading on set theory and beside of my question I will be glad for a recommendation for a good book that explains well cardinal numbers arithmetic.
If we define that $\kappa $ is a cardinal number if $\kappa$ is an ordinal number such that for every $\alpha <\kappa$ there is no $f:\alpha\to\kappa$ that is surjective, in what sense for two infinite cardinal numbers $\kappa$ and $\gamma$ , we have that $\kappa +\gamma=\kappa\cdot\gamma=\max \{\kappa,\gamma\}$? Is this ordinal arithmetic? or is it arithmetic in the sense of bijections of disjoint unions and products?
This is cardinal arithmetic, not ordinal arithmetic, so yes, it's all about existence of bijections related to disjoint unions and products.
One additional comment: $\kappa \cdot \gamma = \max \{ \kappa,\gamma \}$ requires choice to be proven for general infinite cardinals.