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For any hard partition of states, we can define the concept of lumpability. Lumpability is a requirement of clustering. This concept first appeared in Kemeny, Snell's Finite Markov Chains in 1969 <ref name=":33">Kemeny, John G., and J. Laurie Snell. ''Finite markov chains''. Vol. 26. Princeton, NJ: van Nostrand, 1969. https://www.math.pku.edu.cn/teachers/yaoy/Fall2011/Kemeny-Snell_Chapter6.3-4.pdf</ref>. Lumpability is a mathematical condition to determine whether a hard-blocked microscopic state grouping scheme is reducible to the macroscopic TPM. No matter which hard-blocking scheme the state space is partitioned according to, it has a corresponding coarse-graining scheme for the TPM and distributions <ref>Buchholz, Peter. "Exact and ordinary lumpability in finite Markov chains." ''Journal of applied probability'' 31.1 (1994): 59-75.</ref>.

For any hard partition of states, we can define the concept of lumpability. Lumpability is a requirement of clustering. This concept first appeared in Kemeny, Snell's Finite Markov Chains in 1969 <ref name=":33">Kemeny, John G., and J. Laurie Snell. ''Finite markov chains''. Vol. 26. Princeton, NJ: van Nostrand, 1969. https://www.math.pku.edu.cn/teachers/yaoy/Fall2011/Kemeny-Snell_Chapter6.3-4.pdf</ref>. Lumpability is a mathematical condition to determine whether a hard-blocked microscopic state grouping scheme is reducible to the macroscopic TPM. No matter which hard-blocking scheme the state space is partitioned according to, it has a corresponding coarse-graining scheme for the TPM and distributions <ref>Buchholz, Peter. "Exact and ordinary lumpability in finite Markov chains." ''Journal of applied probability'' 31.1 (1994): 59-75.</ref>.

Suppose a grouping method '''<math>S={\A_1\cup A_2\cup ... ,A_r\}</math>''' is given for the state space '''<math>A</math>'''. Here [math]A_i[/math] is any subset of the state space '''<math>S</math>''' and satisfies [math]A_i\cap A_j= \Phi[/math], where [math]\Phi[/math] represents the empty set. [math]\displaystyle{ s^{(t)} }[/math] represents the microscopic state of the system at time [math]\displaystyle{ t }[/math]. The microscopic state space is [math]\displaystyle{ S=\{s_1, s_2,...,s_n\} }[/math] and the microscopic state '''<math>s_i\in S</math>''' are all independent elements in the Markov state space. Let the transition probability from microscopic state <math>s_k</math> to <math>s_m</math> be <math>p_{s_k \rightarrow s_m} = p(s^{(t)} = s_m | s^{(t-1)} = s_k)</math>, and the transition probability from microscopic state <math>s_k</math> to macroscopic state <math>A_i</math> be <math>p_{s_k \rightarrow A_i} = p(s^{(t)} \in A_i | s^{(t-1)} = s_k)</math>. Then the necessary and sufficient condition for lumpability is that for any pair <math>A_i, A_j</math>, <math>p_{s_k \rightarrow A_j}</math> of every state <math>s_k</math> belonging to <math>A_i</math> is equal, that is:

Suppose a grouping method '''<math>S=\{A_1\cup A_2\cup ... ,A_r\}</math>''' is given for the state space '''<math>A</math>'''. Here [math]A_i[/math] is any subset of the state space '''<math>S</math>''' and satisfies [math]A_i\cap A_j= \Phi[/math], where [math]\Phi[/math] represents the empty set. [math]\displaystyle{ s^{(t)} }[/math] represents the microscopic state of the system at time [math]\displaystyle{ t }[/math]. The microscopic state space is [math]\displaystyle{ S=\{s_1, s_2,...,s_n\} }[/math] and the microscopic state '''<math>s_i\in S</math>''' are all independent elements in the Markov state space. Let the transition probability from microscopic state <math>s_k</math> to <math>s_m</math> be <math>p_{s_k \rightarrow s_m} = p(s^{(t)} = s_m | s^{(t-1)} = s_k)</math>, and the transition probability from microscopic state <math>s_k</math> to macroscopic state <math>A_i</math> be <math>p_{s_k \rightarrow A_i} = p(s^{(t)} \in A_i | s^{(t-1)} = s_k)</math>. Then the necessary and sufficient condition for lumpability is that for any pair <math>A_i, A_j</math>, <math>p_{s_k \rightarrow A_j}</math> of every state <math>s_k</math> belonging to <math>A_i</math> is equal, that is:

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