Rosetta biogroupoids RosettaGroupoids 2008-07-12 11:46:26 2008-09-17 01:54:33 Topic topology of social interactions social networking organsimic Supercategories ultra-complex systems % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these 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\newcommand{\medn}{\medbreak \noindent} \newcommand{\bign}{\bigbreak \noindent} \newcommand{\lra}{{\longrightarrow}} \newcommand{\ra}{{\rightarrow}} \newcommand{\rat}{{\rightarrowtail}} \newcommand{\ovset}[1]{\overset {#1}{\ra}} \newcommand{\ovsetl}[1]{\overset {#1}{\lra}} \newcommand{\hr}{{\hookrightarrow}} \newcommand{\<}{{\langle}} %\usepackage{geometry, amsmath,amssymb,latexsym,enumerate} %\usepackage{xypic} \def\baselinestretch{1.1} \hyphenation{prod-ucts} %\geometry{textwidth= 16 cm, textheight=21 cm} \newcommand{\sqdiagram}[9]{$$ \diagram #1 \rto^{#2} \dto_{#4}& #3 \dto^{#5} \\ #6 \rto_{#7} & #8 \enddiagram \eqno{\mbox{#9}}$$ } \def\C{C^{\ast}} \newcommand{\labto}[1]{\stackrel{#1}{\longrightarrow}} %\newenvironment{proof}{\noindent {\bf Proof} }{ \hfill $\Box$ %{\mbox{}} \newcommand{\quadr}[4]{\begin{pmatrix} & #1& \\[-1.1ex] #2 & & #3\\[-1.1ex]& #4& \end{pmatrix}} \def\D{\mathsf{D}} \textbf{Background} It seems that the awareness of the self of other individuals developed at first, and then, through--and as \emph{as an extension of the others}-- to oneself, \emph{self awareness} emerged in a final step. Such pre--historic, societal/social interactions that are based on consensus, are also called `mutual'. These considerations lead to a natural representation of one's `self' emergence in terms of the following mathematical concept. \begin{definition} A \emph{Rosetta biogroupoid}, $\mathsf{R}_{\grp}$, is a representation of characteristic, star-connected patterns of two--way social (cooperative) interactions: $\Longleftrightarrow$, or $\Updownarrow$, that are defined as topological groupoids $\grp_S$ with star-- ($*$), or rose-- like, internal symmetries. From a strictly topological viewpoint, a \emph{Rosetta biogroupoid} is a particular type of highly connected network, or oriented multi--graph, consisting of central nodes (vertices) connected through bi-directional edges representing the mutual interactions between individuals in a population or society; thus, it can also be regarded as a particular type of low-dimensional \PMlinkname{CW complex with a (CW) cellular structure}{CWComplexDefinitionRelatedToSpinNetworksAndSpinFoams} determined by the organisms, such as humans, and their mutual interactions. \end{definition} \textbf{Example} An especially interesting case of Rosetta groupoids is that of \emph{freely generated structures over a graph}, or \emph{network}, of social interactions with different degrees/orders or levels of connectivities; a good example is that of the Erd\"os `connectivity' number, defined by the length of the chain connecting an Erd\"os co-author of a published mathematical paper with Erd\"os to another co-author of the first one, and so on. A \emph{`Rosetta biogroupoid'} structure can be depicted as in the following diagram, but possibly with as many as twenty five branches from the center, reference individual: \begin{equation} \xymatrix{ & \text{Neighbour's Self} \ar[d] & \\ \text{Neighbour's Self} \ar[r] & {\text{Oneself}} \ar[r] \ar[l] \ar[u]\ar[d] & \text{Neighbour's Self} \ar[l] \\ & \text{Neighbour's Self} \ar[u] & } \end{equation} %.............................$Neighbor's.. SELF$ %\med %...........................................$\Updownarrow$ %\med %$Neighbor's.SELF \Longleftrightarrow 1-SELF\Longleftrightarrow Neighbor's. SELF$ %\med...........................................$\Updownarrow$ %\med %$.............................$Neighbor's.. SELF$ \med \noindent {\textbf{Diagram 1}}: \emph{A Rosetta biogroupoid} of consensual, societal interactions leading to self-awareness, one's self and full consciousness; there could be as few as five, or as many as twenty five, individuals in a pre-historic society of humans; here only four are represented as branches. \textbf{Remark:} such cooperative interactions caused- through social and biological coevolution- the emergence of human (and/or `humanoid') structured languages with both syntax and semantics; the latter in their turn facilitated the development of self-awareness and the emergence of the human mind as an ultra-complex system, and consciousness as an ultra-complex (meta-process) of processes.