A probabilistic and algebraic treatment of regular inbreeding systems

Summary A probabilistic and algebraic treatment of regular inbreeding systems is presented. Regular inbreeding systems can be thought of as graphs which have certain natural homogeneity properties. Random walks X_n and Y_n are introduced on the nodes of the graphs; the event {X_n = Y_n} is a renewal event by the homogeneity property. We show that in such regular inbreeding systems the population becomes genetically uniform if and only if the event {X_n = Y_n} is recurrent, which happens if 1/ A_n diverges, where A_n is the number of ancestors n generations into the past. We give two counterexamples to show the converse is false in general, but we verify the converse in the case of the graphs of certain finitely presented semigroups.An expended version of this paper was submitted as a doctoral thesis to Purdue University. This thesis was directed by Professor Stanley Sawyer.