Evolution Algebras and Markov Chains

For a Markov chain, we can define an evolution algebra by taking states as generators and transition probability vectors as defining relations. We may say an evolution algebra defined by a Markov chain is a Markov evolution algebra. Every property of a Markov chain can be redefined by its Markov evolution algebra. In other words, properties of Markov chains can be revealed by studying their evolution algebras. Moreover, Markov chains, as a type of dynamical systems, have a hidden algebraic aspect. In first three sections of this chapter we study the relations between Markov chains and evolution algebras. In the last section, the hierarchy of a general Markov chain is revealed naturally by its evolution algebra.