The Algebraic Structure of Physical Quantities

Abstract

We present a general algebraic basis for arbitrary systems of units such as those used in physical sciences, engineering, and economics. Physical quantities are represented as q-numbers: an ordered pair u = {u,label_u}, that is, u ∈ q = X × W _B . The algebraic structure of the infinite sets of labels that represent the “units” has been established: such sets W _B are infinite Abelian multiplicative groups with a finite basis. W _B is solvable as it admits a tower of Abelian subgroups. Extensions to include the possibility of rational powers of labels have been included, as well as the addition of named labels. Named labels are an essential feature of all practical systems of units. Furthermore, q is an Abelian multiplicative group, and it is not a ring. q admits decomposition into one-dimensional normed vector spaces over the field X among members with equivalent labels. These properties lead naturally to the concept of well-posed relations, and to Buckingham’s theorem of dimensional analysis. Finally, a connection is made with a Group Ring structure and an interpretation in terms of the observable properties of physico-chemical systems is given.