"$\leq_\Bbb{R}$" is restriction of "$\leq_{\overline{\Bbb{R}}}$" to $\Bbb{R}$ ?!

Question

In definition of Affinely Extended Real Numbers, I think that "$\leq_\Bbb{R}$" is restriction of "$\leq_{\overline{\Bbb{R}}}$" to $\Bbb{R}$

Is it correct?... Thanks in advance!

Answer 1

We can define $\le_{\overline{\mathbb R}}\;\subseteq \overline{\mathbb R}\times\overline{\mathbb R}$ as $$ \le_{\overline{\mathbb R}}\quad=\quad\le_{\mathbb R}\ \cup\ \left\{\, (-\infty, x) \ \middle|\ x\in\overline{\mathbb R}\,\right\}\ \cup\ \left\{\, (x, \infty) \ \middle|\ x\in\overline{\mathbb R}\,\right\}. $$

Then $$\le_{\overline{\mathbb R}}\ \cap\ (\mathbb R\times\mathbb R)\quad=\quad \le_{\mathbb R}.$$