# Table 2 Conditions required for a 4-state Markovian process to be lumpable (in terms of s and π), and transformations to obtain $\stackrel{˜}{\pi }$ and $\stackrel{˜}{s}$ such that the lumpability holds.

${\mathcal{S}}^{\prime }$ Lumpability conditions ($\stackrel{˜}{\pi }$,$\stackrel{˜}{s}$)
{{A, G}, C, T} s12 = s23
s14 = s34
${\stackrel{˜}{s}}_{\mathsf{\text{13}}}=\phantom{\rule{0.5em}{0ex}}{\stackrel{˜}{s}}_{\mathsf{\text{23}}}=\left({ŝ}_{\mathsf{\text{13}}}+{ŝ}_{\mathsf{\text{23}}}\right)/\mathsf{\text{2}}$
${\stackrel{˜}{s}}_{\mathsf{\text{14}}}=\phantom{\rule{0.5em}{0ex}}{\stackrel{˜}{s}}_{\mathsf{\text{34}}}=\left({ŝ}_{\mathsf{\text{14}}}+{ŝ}_{\mathsf{\text{34}}}\right)/\mathsf{\text{2}}$
{A, G, {C, T}} s12 = s14
s23 = s34
${\stackrel{˜}{s}}_{\mathsf{\text{12}}}=\phantom{\rule{0.5em}{0ex}}{\stackrel{˜}{s}}_{14}=\left({ŝ}_{\mathsf{\text{12}}}+{ŝ}_{\mathsf{\text{14}}}\right)/\mathsf{\text{2}}$
${\stackrel{˜}{s}}_{\mathsf{\text{23}}}=\phantom{\rule{0.5em}{0ex}}{\stackrel{˜}{s}}_{\mathsf{\text{34}}}=\left({ŝ}_{\mathsf{\text{23}}}+{ŝ}_{\mathsf{\text{34}}}\right)/\mathsf{\text{2}}$
{A, {C, G}, T} s12 = s13
s24 = s34
${\stackrel{˜}{s}}_{\mathsf{\text{12}}}=\phantom{\rule{0.5em}{0ex}}{\stackrel{˜}{s}}_{\mathsf{\text{13}}}=\left({ŝ}_{\mathsf{\text{12}}}+{ŝ}_{\mathsf{\text{13}}}\right)/\mathsf{\text{2}}$
${\stackrel{˜}{s}}_{\mathsf{\text{24}}}=\phantom{\rule{0.5em}{0ex}}{\stackrel{˜}{s}}_{\mathsf{\text{34}}}=\left({ŝ}_{\mathsf{\text{24}}}+{ŝ}_{\mathsf{\text{34}}}\right)/\mathsf{\text{2}}$
{C, G, {A, T}} s12 = s24
s13 = s34
${\stackrel{˜}{s}}_{\mathsf{\text{12}}}=\phantom{\rule{0.5em}{0ex}}{\stackrel{˜}{s}}_{\mathsf{\text{24}}}=\left({ŝ}_{\mathsf{\text{12}}}+{ŝ}_{\mathsf{\text{24}}}\right)/\mathsf{\text{2}}$
${\stackrel{˜}{s}}_{\mathsf{\text{13}}}=\phantom{\rule{0.5em}{0ex}}{\stackrel{˜}{s}}_{\mathsf{\text{34}}}=\left({ŝ}_{\mathsf{\text{13}}}+{ŝ}_{\mathsf{\text{34}}}\right)/\mathsf{\text{2}}$
{{A, C}, G, T} s13 = s23
s14 = s24
${\stackrel{˜}{s}}_{\mathsf{\text{13}}}=\phantom{\rule{0.5em}{0ex}}{\stackrel{˜}{s}}_{\mathsf{\text{23}}}=\left({ŝ}_{\mathsf{\text{13}}}+{ŝ}_{\mathsf{\text{23}}}\right)/\mathsf{\text{2}}$
${\stackrel{˜}{s}}_{\mathsf{\text{14}}}=\phantom{\rule{0.5em}{0ex}}{\stackrel{˜}{s}}_{\mathsf{\text{24}}}=\left({ŝ}_{\mathsf{\text{14}}}+{ŝ}_{\mathsf{\text{24}}}\right)/\mathsf{\text{2}}$
{A, C, {G, T}} s13 = s14
s12 = s13
${\stackrel{˜}{s}}_{\mathsf{\text{13}}}=\phantom{\rule{0.5em}{0ex}}{\stackrel{˜}{s}}_{\mathsf{\text{14}}}=\left({ŝ}_{\mathsf{\text{13}}}+{ŝ}_{\mathsf{\text{14}}}\right)/\mathsf{\text{2}}$
${\stackrel{˜}{s}}_{\mathsf{\text{23}}}={\stackrel{˜}{s}}_{\mathsf{\text{24}}}=\left({ŝ}_{\mathsf{\text{23}}}+{ŝ}_{\mathsf{\text{24}}}\right)/\mathsf{\text{2}}$
{A, {C, G, T}} s12 = s13
s13 = s14
${\stackrel{˜}{s}}_{\mathsf{\text{12}}}={\stackrel{˜}{s}}_{\mathsf{\text{13}}}=\phantom{\rule{0.5em}{0ex}}{\stackrel{˜}{s}}_{\mathsf{\text{14}}}=\left({ŝ}_{\mathsf{\text{12}}}+{ŝ}_{\mathsf{\text{13}}}+{ŝ}_{\mathsf{\text{14}}}\right)/\mathsf{\text{3}}$
{C, {A, G, T}} s12 = s23
s23 = s24
${\stackrel{˜}{s}}_{\mathsf{\text{12}}}={\stackrel{˜}{s}}_{\mathsf{\text{23}}}=\phantom{\rule{0.5em}{0ex}}{\stackrel{˜}{s}}_{\mathsf{\text{24}}}=\left({ŝ}_{\mathsf{\text{12}}}+{ŝ}_{\mathsf{\text{23}}}+{ŝ}_{\mathsf{\text{24}}}\right)/\mathsf{\text{3}}$
{G, {A, C, T}} s13 = s23
s23 = s34
${\stackrel{˜}{s}}_{\mathsf{\text{13}}}={\stackrel{˜}{s}}_{\mathsf{\text{23}}}=\phantom{\rule{0.5em}{0ex}}{\stackrel{˜}{s}}_{\mathsf{\text{34}}}=\left({ŝ}_{\mathsf{\text{13}}}+{ŝ}_{\mathsf{\text{23}}}+{ŝ}_{\mathsf{\text{34}}}\right)/\mathsf{\text{3}}$
{T, {A, C, G}} s14 = s24
s24 = s34
${\stackrel{˜}{s}}_{\mathsf{\text{14}}}={\stackrel{˜}{s}}_{\mathsf{\text{24}}}=\phantom{\rule{0.5em}{0ex}}{\stackrel{˜}{s}}_{\mathsf{\text{34}}}=\left({ŝ}_{\mathsf{\text{14}}}+{ŝ}_{\mathsf{\text{24}}}+{ŝ}_{\mathsf{\text{34}}}\right)/\mathsf{\text{3}}$
{{A, G}, {C, T}} s12π2 + s14π4 = s23π2+ s34π4
s12π1 + s23π3 = s14π1+ s34π3
${\stackrel{˜}{s}}_{23}=\frac{{\stackrel{˜}{s}}_{12}\left({\stackrel{^}{\pi }}_{2}{\stackrel{^}{\pi }}_{3}-{\stackrel{^}{\pi }}_{1}{\stackrel{^}{\pi }}_{4}\right)\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{\stackrel{˜}{s}}_{14}{\stackrel{^}{\pi }}_{4}\left({\stackrel{^}{\pi }}_{1}+{\stackrel{^}{\pi }}_{3}\right)\phantom{\rule{0.3em}{0ex}}}{{\stackrel{^}{\pi }}_{3}\left({\stackrel{^}{\pi }}_{2}+{\stackrel{^}{\pi }}_{4}\right)\phantom{\rule{0.3em}{0ex}}}$
${\stackrel{˜}{s}}_{34}=\frac{{\stackrel{˜}{s}}_{12}{\stackrel{^}{\pi }}_{1}+{\stackrel{˜}{s}}_{23}{\stackrel{^}{\pi }}_{3}-{\stackrel{˜}{s}}_{14}{\stackrel{^}{\pi }}_{1}}{{\stackrel{^}{\pi }}_{3}}$
{{A, T}, {C, G}} s12π2 + s13π3 = s24π2+ s34π3
s12π1 + s24π4 = s13π1+ s34π4
${\stackrel{˜}{s}}_{13}=\frac{{\stackrel{˜}{s}}_{12}\left({\stackrel{^}{\pi }}_{1}{\stackrel{^}{\pi }}_{3}-{\stackrel{^}{\pi }}_{2}{\stackrel{^}{\pi }}_{4}\right)\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{\stackrel{˜}{s}}_{24}{\stackrel{^}{\pi }}_{4}\left({\stackrel{^}{\pi }}_{2}+{\stackrel{^}{\pi }}_{3}\right)\phantom{\rule{0.3em}{0ex}}}{{\stackrel{^}{\pi }}_{3}\left({\stackrel{^}{\pi }}_{1}+{\stackrel{^}{\pi }}_{4}\right)\phantom{\rule{0.3em}{0ex}}}$
${\stackrel{˜}{s}}_{34}=\frac{{\stackrel{˜}{s}}_{12}{\stackrel{^}{\pi }}_{2}+{\stackrel{˜}{s}}_{13}{\stackrel{^}{\pi }}_{3}-{\stackrel{˜}{s}}_{24}{\stackrel{^}{\pi }}_{2}}{{\stackrel{^}{\pi }}_{3}}$
{{A, C}, {G, T}} s13π3 + s14π4 = s23π3 + s24π4
s13π1 + s23π2 = s14π1 + s24π4
${\stackrel{˜}{s}}_{23}=\frac{{\stackrel{˜}{s}}_{13}\left({\stackrel{^}{\pi }}_{2}{\stackrel{^}{\pi }}_{3}-{\stackrel{^}{\pi }}_{1}{\stackrel{^}{\pi }}_{4}\right)\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{\stackrel{˜}{s}}_{14}{\stackrel{^}{\pi }}_{4}\left({\stackrel{^}{\pi }}_{1}+{\stackrel{^}{\pi }}_{2}\right)\phantom{\rule{0.3em}{0ex}}}{{\stackrel{^}{\pi }}_{2}\left({\stackrel{^}{\pi }}_{3}+{\stackrel{^}{\pi }}_{4}\right)\phantom{\rule{0.3em}{0ex}}}$
${\stackrel{˜}{s}}_{24}=\frac{{\stackrel{˜}{s}}_{13}{\stackrel{^}{\pi }}_{1}+{\stackrel{˜}{s}}_{23}{\stackrel{^}{\pi }}_{2}-{\stackrel{˜}{s}}_{14}{\stackrel{^}{\pi }}_{1}}{{\stackrel{^}{\pi }}_{2}}$