Thus a, b ∈ A$_{i}$ satisfy the conditions in step 2 for construction B. This contradicts our assumption that we used construction A, so there is an arrow formed by x$_{i}$ and another vertex of D$_{i}$$_{+1}$ . In construction B we have b$_{i}$y$_{i}$ ∈ A ( D$_{i}$ ) so b$_{i}$z / ∈ T$_{i}$ for all z ∈ V ( D$_{i}$ ) \ { x$_{i}$ } . This gives b$_{i}$x$_{i}$ ∈ σ$_{i}$ and b$_{i}$x$_{i}$ ∈ A ( D$_{i}$$_{+1}$ ) so there is an arrow formed by x$_{i}$ with another vertex of D$_{i}$$_{+1}$ . Acknowledgements The author wishes to thank Martin Erickson of Truman State University and Steven Szydlik of the University of Wisconsin Oshkosh for encouragement and suggesting improvements to the writing in earlier drafts. References [1] G. Abrams, Groups Gradings and Recovery Results for Generalized Incidence Rings, J. Alg. 164 (1994) 859-875. [2] K. Kuratowski, Sur le probl´ eme des courbes gauches en topologie, Fund. Math. 15 (1930) 271–283. [3] K. Price, Good Gradings of Generalized Incidence Rings, submitted [4] E. Spiegel and C. O’Donnell, ”Incidence Algebras,” Marcel Dekker, New York, NY (1997). 18