24 
INTRODUCTION, DEFINITIONS 
[chap. I 
But under the definition in § 4, no specific set of 
basal units is implied,* and we obtain the same algebra 
(not merely an equivalent one) when we make a trans 
formation of units with coefficients in F. That definition 
by wholly undefined elements is well adapted to the 
treatment of difference algebras (§ 25) which are abstract 
algebras whose elements are certain classes of things. 
The same definition without postulate V is convenient 
in the study of algebras of infinite order (not treated in 
this book), an example being the field of all real numbers 
regarded as an algebra over the field of rational numbers. 
* To emphasize this point, we may understand postulate V (that a 
finite; basis exists) to mean that there is an upper limit to the number of 
linearly independent elements which can be chosen in the algebra.