Rosetta biogroupoids

BackgroundIt seems that the awareness of the self of other individuals developed at first, and then,through–and as as an extension of the others– to oneself, 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.

Definition 0.1.

A Rosetta biogroupoid, $\\mathsf{R}_{{\\mathsf{G}}}$ , is a representation of characteristic, star-connectedpatterns of two–way social (cooperative) interactions: $\\Longleftrightarrow$ , or $\\Updownarrow$ , that are defined as topological groupoids ${\\mathsf{G}}_{S}$ with star– ( $*$ ), or rose– like, internal symmetries. From a strictly topologicalviewpoint, a 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 interactionsbetween individuals in a population or society; thus, it can also be regarded as a particular type of low-dimensionalCW complex with a (CW) cellular structure (http://planetmath.org/CWComplexDefinitionRelatedToSpinNetworksAndSpinFoams) determined by the organisms, such as humans, and their mutual interactions.

Example An especially interesting case of Rosetta groupoids is that of freely generated structures over a graph, or network, of social interactions with different degrees/orders or levels of connectivities; a good example is that of the Erdös ‘connectivity’ number, defined by the length of the chain connecting an Erdös co-author of a published mathematical paper with Erdös to another co-author of the first one, and so on.

A ‘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:

\\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]&}

(0.1)

Diagram 1: A Rosetta biogroupoid ofconsensual, societal interactions leading to self-awareness, one’sself and full consciousness; there could be as few as five, or asmany as twenty five, individuals in a pre-historic society ofhumans; here only four are represented as branches.

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.