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%%% Primary Title: biogroupoids and mathematical models of species evolution
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%%% Filename: BiogroupoidsAndMathematicalModelsOfSpeciesEvolution.tex
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\begin{document}
\subsection{Biogroupoids and mathematical models of species evolution}
\subsubsection{Introduction}
\emph{Biogroupoids}, $\mathcal{G_B}$, were introduced as mathematical \emph{\htmladdnormallink{representations}{http://planetphysics.us/encyclopedia/CategoricalGroupRepresentation.html} of
evolving biological species} (\cite{BBGG2k6, BBG2k7}) that are defined by (or `consist of')
\emph{weakly equivalent classes of living organisms}, $E_O$, specified by inter-breeding organisms;in this case, the
\emph{weak \htmladdnormallink{equivalence relation}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}}, $\sim_w$, is defined on the set of evolving organisms modeled in terms of functional,
\emph{isomorphic genome networks}, $G_{iso}^N$, such as those described by $LM_n$-logic networks in \L{}ukasiewicz-Moisil, $\mathcal{L}M$ topoi (\cite{BBGG2k6}).
\subsection{AT-Formulation}
This biogroupoid \htmladdnormallink{concept}{http://planetphysics.us/encyclopedia/PreciseIdea.html} allows an \htmladdnormallink{algebraic topology}{http://planetphysics.us/encyclopedia/CubicalHigherHomotopyGroupoid.html} formulation of the origin of species and biological evolution
both at organismal/organismic and biomolecular levels; it represents a new approach to biological evolution
from the standpoint of super-complex \htmladdnormallink{systems biology}{http://planetphysics.us/encyclopedia/SystemsBiology.html}.
\begin{thebibliography}{9}
\bibitem{BBGG2k6}
Baianu I. C., Brown R., Georgescu G. and J. F. Glazebrook: 2006, Complex Nonlinear Biodynamics in Categories, Higher Dimensional Algebra and \L ukasiewicz-Moisil Topos: Transformations of Neuronal, Genetic and Neoplastic Networks., \emph{Axiomathes}, \textbf{16} Nos. 1-2: 65-122.
\bibitem{BBG2k7}
Baianu, I.C., R. Brown and J.F. Glazebrook. : 2007, Categorical Ontology of Complex Spacetime Structures: The Emergence of Life and Human Consciousness, Axiomathes, \textbf {17}: 35-168.
\end{thebibliography}
\end{document}
