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On real chains of evolution algebras. (English) Zbl 1361.17030

Summary: We define a chain of \(n\)-dimensional evolution algebras corresponding to a permutation of \(n\) numbers. We show that a chain of evolution algebras (CEA) corresponding to a permutation is non-trivial if and only if the permutation has a fixed point. We show that a CEA is a chain of nilpotent algebras (independently on time) if it is trivial. We construct a wide class of chains of three-dimensional EAs and a class of symmetric \(n\)-dimensional CEAs. A construction of arbitrary dimensional CEAs is given. Moreover, for a chain of three-dimensional EAs, we study the behaviour of the baric property, the behaviour of the set of absolute nilpotent elements and dynamics of the set of idempotent elements depending on the time.

MSC:

17D92 Genetic algebras
17D99 Other nonassociative rings and algebras
60J27 Continuous-time Markov processes on discrete state spaces
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