A physical dimension is a property we associate with physical quantities for purposes of classification or differentiation. Mass, length, and force are examples of physical dimensions. Composite physical dimensions can be described by composing primitive dimensions. For example, Length/Time ('length over time') is a dimension that can be associated with a velocity.
The composition operators for dimensions are * [dimension product] and expt [exponentiation to a real power], which have algebraic properties analogous to their use with real numbers. The product of any two dimensions is a dimension. There is an indentity element for * on dimensions; it is called the identity-dimension. The product of any dimension and the identity-dimension is the original dimension; any other product defines a new dimension. The analogy of division is exponentiation to a negative number.
There is no intrinsic property of a dimension that makes it primitive. A set of primitive dimensions is chosen by convention to define a system of units of measure. However, the relative relationships among dimensions can be established independently of systems of units. For example, the dimension corresponding to velocity is length/time, and therefore the length dimension is the same as velocity * time. This is true regardless of whether the length or velocity dimensions are viewed as the fundamental dimensions in some system, or whether either dimension is denoted by a object constant or a term expression in some theory.
(Abelian-Group Physical-Dimension * Identity-Dimension)