Notation: I use " +' ", " -' ", " x' ", and " /' " to represent the four arithmetic operations.

I interpret your definition of +' as: a +' b = append b items to an a-list and count interstices. (N.B. it's an operator definition, and that the identity element for +' is 1.)

1.

If -' is defined as a -' b = remove b items from an a-list and count interstices (i.e. the logical complement of +'), then

If a -' b is defined as the inverse of the +' operation, such that a +' b -' b = a = a -' b +' b, then

(Using Common Notion 2: "If equals be added to equals, the wholes are equal.")

Note that the second comports with the notion that a -' a should equal the identity element: 1. (This is why I favor the second.)

2.

If x' is defined as repeated addition (via +'), such that 2 x' 4 = 4 +'4, then

Note that the ordering is that implied by the old-fashioned reading of 2 x 4 as "twice four," or "two fours." This definition has the interesting, and possible undesirable consequence that 1 is a left-identity but not a right-identity: 1 x' 4 = 4, but 4 x' 1 = 3. (All multiplications not containing a one commute.)

Lack of universal commutivity can be remedied by defining 1 x' 4 as 4 +' 0; i.e. multiplying by one is adding nothing to 4, which then requires the final counting of interstices. In this case 1 x' 4 = 3 = 4 x' 1, all multiplication commutes, but there is no multiplicative identity.

I'd be interested to hear opinions on which definition of 1 x' a is preferable. The second, containing some appeal, leads to a breakdown in distributivity when 1's are involved. The first, I believe, allows for distributivity in all cases. (I've not rigorously proven it, though.)

3.

Properly speaking, I'd say

However, we don't really want that result, do we? So, let's clean up our definitions by dropping the notion of items-and-spaces, and just go with

a +' b = a + b -1 (where " + " is the usual addition operator),

a x' b = a x b -1 (ditto).

Note that a +' 1 = a, and a +' b = b +' a. Good so far.

a x' b = b x' a, but we still have no (and have no hope of a) multiplicitave identity element. This has grave consequences for the idea of division, but we forge ahead blindly...

Defining division as the inverse of the multiplication operation, such that a x' b /' b = a = a /' b x' b, we can use the above to determine that

4.

Using the most-recent definition of " x' ", and assuming that sqrt'(a) is the inverse of a x' a, we see that