Category Theory for Programmers Chapter 24: F-Algebras

14 Dec 2018

1. Implement the evaluation function for a ring of polynomials of one variable.

   For a ring $\mathscr{R}$, then define

   \[\mathscr{R}\left[t\right] = \left\{ \sum^n_i a_i t^i \,\middle|\, a_i \in \mathscr{R} \right\}\]

   Then we can define the following:

   • $0{\mathscr{R}\left[t\right]} = 0{\mathscr{R}} t^0$
   • $1{\mathscr{R}\left[t\right]} = 1{\mathscr{R}} t^0$
   • for $\left(\sum a_i t^i\right), \left(\sum b_i t^i\right) \in \mathscr{R}\left[t\right]$, we have
     • $\left(\sum a_i t^i\right) +_{\mathscr{R}\left[t\right]} \left(\sum b_i t^i\right) = \sum (a_i + b_i) t^i$
     • $\left(\sum a_i t^i\right) \times_{\mathscr{R}\left[t\right]} \left(\sum b_i t^i\right) = \sum \left(\sum_{x+y=i} a_x b_y\right) t^i$

2. Generalize previous construction to polynomials of many independent variables.

   Since $\mathscr{R}[t]$ is a ring, so too is $(\mathscr{R}[t])[u]$. In this way, $\mathscr{R}[t_0, …, t_n]$ can be defined as $\mathscr{R}[t_0][…][t_n]$.