On the Algebra of Lists #

In this article, we derive a formal algebra for lists, based on the existing algebra of sets. We develop basic properties, laws, and operations on lists such as equivalence, shift, slice, sublist, dot product, and concatenation. We then proceed to demonstrate the applications of the algebra in proving the existence and uniqueness of prime factorization, and the base-$b$ representation of natural numbers. We then go on to demonstrate the building of a definition for polynomials using list-algebra, which allows us to prove some well known results on polynomials rigorously.