  Genetic and evolution algebras arise naturally from applied probability and
stochastic processes. Gibbs measures describe interacting systems commonly
studied in thermodynamics and statistical mechanics with applications in
several fields. Here, we consider that the algebras are determined by
configurations of finite spins on a countable set with their associated Gibbs
distributions. The model preserves properties of the finite-dimensional Gibbs
algebras found in the literature and extend their results. We introduce
infertility in the genetic dynamics when the configurations differ
macroscopically. It induces a decomposition of the algebra into a direct sum of
fertile ideals with genetic realization.
  The proposed infinite-dimensional algebras are commutative, non-associative,
with uncountable basis and zero divisors. The properties of Gibbs measures
allow us to deal with the difficulties arising from the algebraic structure and
obtain the results presented in this article.
