An Isomorphism Between the P-adic Integers and a Ring Associated with a Tiling of N-space By Permutohedra

The classical lattice A ,*, whose Voronoi cells tile Euclidean n-space by permutohedra, can be given the generalized balance ternary ring structure GBT, in a natural way as a quotient ring of Z [xl. The ring GBT, can also be considered as the set of all finite sequences sO sl.. . sk, with si E GBT,/srGBT, for all i, where M is an appropriately chosen element in GBT,. The extended generalized balance ternary (EGBT,) ring consists of all such infinite sequences. A primary goal of this paper is to prove that if 2”+ ’ 1 and n + 1 are relatively prime, then EGBT, is isomorphic as a ring to the (2”+’ 1)-adic integers.