ABSTRACT
In a first course in linear algebra, a rank function is usually defined as the dimension of the row space of a real n × n matrix or the number of nonzero rows in a row echelon form of the matrix. Here, we shall use a general definition of a rank function, incorporating its most basic properties, that being that a rank function is a mapping from an algebraic system with a binary operation (we shall call it addition) to the set of nonnegative integers which must map only the additive identity to 0, and be subadditive, that is the “rank” of a sum must be at most the sum of the “ranks” of the summands. Formally:
Let N https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429280092/903cf4ed-cbd9-4ecc-bd90-95916cd1752b/content/inline-math14_1.jpg"/> denote the semiring of natural numbers (nonnegative integers), and let Q https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429280092/903cf4ed-cbd9-4ecc-bd90-95916cd1752b/content/inline-math14_2.jpg"/> be an additive Abelian monoid with identity O. Then a mapping f : Q → N https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429280092/903cf4ed-cbd9-4ecc-bd90-95916cd1752b/content/inline-math14_3.jpg"/> is a rank function if for A , B ∈ Q https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429280092/903cf4ed-cbd9-4ecc-bd90-95916cd1752b/content/inline-math14_4.jpg"/> ,
f(A) = 0 if and only if A = O and
f is subadditive, that is f(A + B) ≤ f(A) + f(B).
