Arithmetic on Blocks

We introduce a binary operation on strings (blocks) of elements from the set
{0,1,\dots, m-1}, where m is an arbitrary integer greater than 1. This operation is
an extension of one introduced by Konrad Jacobs and Michael Keane in the 1960's for
blocks of 0's and 1's. We show that the extended operation is associative, introduce the
concept of similar, cyclic, and circular blocks and provide a unique factorization theorem
under this operation up to the similarity of the factors. We also give the conditions for
commutativity of indecomposable blocks.