From what I can tell, his notation, while cumbersome, is an attempt to attach a physical interpretation to the division by zero that is unchanged when viewed through the other binary operators of addition, subtraction, and multiplication across all real numbers. While it is not explicitly stated, it appears that the physical interpretation of division by zero is "leave the original quantity alone" or maybe "don't do the division" since if you divide the quantity into zero groups, you never touched the original quantity. You basically did not do the division. I am in full agreement that the notation is confusing, and I still am not sure I grasp how or why it is necessary. But, suspending my disbelief and pressing onward, the overall gist made more sense than the specifics of the notation. The point of divergence between his theory and more well-known examples of algebras that allow division by zero (such as meadows and wheels) is that he is explicitly not defining $0^{-1}$ because that is where most theories run into problems. Unlike most systems with division by zero, he does not allow $\dfrac{0}{0} = 1$. He has $\dfrac{0}{0} = 0$. Now, if all divisions by zero resulted in zero, that would also cause problems and violate the field axioms, as well (see involutive meadows). But, treating division by zero as multiplication by one seems to avoid many of the algebraic problems that result from defining $0^{-1}$. I have not comprehensively looked into this, but I do not see anything glaringly off that would indicate it would fail. It still might, and I reserve judgment, but it is a novel approach to defining a system with division by zero at the very least.