= Types of pairs of vectors over a finite field
{title2=$|S_2(T)|=q+3,\quad|F_2/E_2(T)|=2^{q+3}$}
For infinite spaces over a <field> of size $q$, two-variable types are determined by the kernel of the <linear map> sending $(a,b)$ to $ax+by$. Its ranks give the zero pair, $q+1$ one-dimensional directions and the independent-pair type. All $q+3$ types are isolated by finite formulas. <Quantifier elimination> makes the <Lindenbaum algebra> their full power-set <Boolean algebra>. The one-variable algebra has just the zero and nonzero <atom of a Boolean algebra>[atoms], hence four <Elements>.
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