[AntonioLao]

An integral domain is a mathematical system to which certain postulates hold exclusively for this particular system in question. For example, the system of positive and negative integers: 0, ±1, ±2, ±3, ±4, ±5, ±6, ±7, ±8, …, to infinity. They hold the following postulates: (1) closure, (2) uniqueness, (3) commutative laws, (4) associative laws, (5) distributive law, (6) zero, (7) unity, ( additive inverse, and (9) cancellation law.

On the other hand, a differential domain is a system of Hadamard matrices. Moreover, the postulates for this system only apply for matrices of equivalent level of existence (LOE). Analogous to the properties held by an abstract ring structure, a differential domain is closed under the operations of addition and multiplications and that is an Abelian group with respect to addition and an associative semigroup with respect to multiplication and in which the distributive laws relating the two operations hold. For matrices A, B, I, and O at LOE=2 where I is the identity matrix and O is the zero matrix, one of the unique properties is AB=2B=B-A if and only if A+B=O. However, for single element matrices A=-1, I=B=1, and O=0 such that AB= -1, while B-A=2=1-(-1) the property does not hold.