Mathematical Appendix

Abstract

Following Weinstein [1] closely, a groupoid, G, is defined by a base set A upon which some mapping—a morphism—can be defined. Note that not all possible pairs of states $(a_{j}, a_{k})$ in the base set A can be connected by such a morphism. Those that can define the groupoid element, a morphism $g=(a_{j},a_{k})$ having the natural inverse $g^{-1}=(a_{k}, a_{j})$. Given such a pairing, it is possible to define “natural” end-point maps $\alpha (g)=a_{j}, \beta (g)=a_{k}$ from the set of morphisms G into A, and a formally associative product in the groupoid $g_{1}g_{2}$ provided $\alpha (g_{1}g_{2})=\alpha (g_{1}), \beta (g_{1}g_{2})=\beta (g_{2})$, and $\beta (g_{1})=\alpha (g_{2})$.