We consider the backward evolution of a particular type of Mendelian genetic system whose transition probabilities give place to the so-called coalgebras with genetic realization and describe the equilibrium states of such mathematical objects and therefore those of the genetic system. We exploit the relationship between the genetic coalgebras modeling the transference of the genetic inheritance and cubic stochastic matrices of type (1, 2) studying first the ergodicity of these matrices in terms of the stationary probability distributions of the bivariate Markov chains defined by their accompanying matrices. Then we apply the obtained results to describe the equilibrium states of coalgebras with genetic realization.
We define a Jordan analogue of Johnson's associative algebra of quotients and study it under a suitable condition of nonsingularity of the Jordan algebra, which we call strong nonsingularity. In particular, we prove the existence and describe the maximal algebras of quotients of prime strongly nonsingular Jordan algebras.
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