The archimedean graph of a positive function on a semigroup

Necessary and sufficient conditions are obtained for the archimedean graph of a positive function on an S-indecomposable semigroup to have no irreducible polygons. Miscellaneous facts about positive functions are obtained. Illustrative examples of positive functions on the free content are given.Some of the results here were announced in Abstract 711-20-44 of Notices AMS. 21 (1974), A-93.