Let $K = \Bbb F_2$. Take four disjoint sets of variables $\Sigma_a$, $\Sigma_b, \Sigma_c, \Sigma_d$, each indexed by $\Bbb N \cup \{ 0 \}$, and denote their union by $\Sigma_0$. Let $A_0 = K\langle \Sigma0 \rangle$ be the free $K$-algebra over these variables.
Set $f(x) = \sum\limits{i = 0}^\infty (a_i + bi x) t^i$, $g(x) = \sum\limits{i = 0}^\infty (c_i + d_i x) t^i \in A_0[[t]][x]$. Take $C_0$ to be the set of $A_0$-coefficients of each monomial in $f(x) g(x)$, and set $B_0 = A_0/(C_0)$ (the quotient by the ideal generated by $C_0$).
Define inductively:
$\Sigma_{i+1} = \Sigma_i \sqcup \{ x_S, y_S: S \subset \Sigma_i, |S| < \infty\}$ (for every finite subset of old variables, adjoin a pair of new variables);
$A{i + 1} = K\langle \Sigma{i + 1} \rangle$;
$C_{i + 1} = C_i \cup \{ x_S a_i, x_S b_i, c_i y_S, d_i y_S, x_S s, s yS: S \in \Sigma{i + 1} \setminus \Sigma_i, s \in S \}$, and
$B{i + 1} = A{i + 1}/(C{i + 1})$.
The ring $R = R{163}$ is the direct limit $\bigcup\limits_{i=0}^\infty B_i$.
Comments
Camillo and Nielsen first set $K$ arbitrary, only to use $\Bbb F_2$ on p602. I think we should specialize to the latter. They also "quotient by a union of ideals" (it seems to mean just an union of their generating sets).
Comments
Camillo and Nielsen first set $K$ arbitrary, only to use $\Bbb F_2$ on p602. I think we should specialize to the latter. They also "quotient by a union of ideals" (it seems to mean just an union of their generating sets).
_Originally posted by @dyunov in https://github.com/rschwiebert/dart_data/issues/40#issuecomment-1902251886_