The formal-power-series Multiplicative Rule of Three

Let $R[[x,y]]$ be the ring of formal power series in $x$ and $y$ with coefficients in a

-algebra $R$. Let $u_1,\dots,u_N,v_1,\dots,v_N\in R$, and suppose that $g_1,\dots,g_N,h_1,\dots,h_N\in R[[x,y]]$ have the forms

g_i=1+\alpha_{i1}xu_i+\alpha_{i2}(xu_i)^2+\alpha_{i3}(xu_i)^3+\cdots,\qquad h_i=1+\beta_{i1}yv_i+\beta_{i2}(yv_i)^2+\beta_{i3}(yv_i)^3+\cdots,

$$with$$

-coefficients $_{ik},_{ik}$, and for every $i$ at least one of $_{i1}$ and $_{i1}$ nonzero. For $S\subset\{1,\dots,N\}$, write $g_S=g_{s_m}\cdots g_{s_1}$ and $h_S=h_{s_m}\cdots h_{s_1}$ when $S=\{s_1<\cdots<s_m\}$. The formal-power-series Multiplicative Rule of Three. The following are equivalent: $g_Sh_S=h_Sg_S$ for every subset $S\subset\{1,\dots,N\}$; and $g_Sh_S=h_Sg_S$ for every subset $S$ of cardinality $1$, $2$, or $3$. Extensive computational evidence supports this assertion, which is presented as an open conjecture after the corresponding purely group-theoretic statement was shown to fail without the formal-power-series hypotheses.