COMMENTS FOR THE AUTHOR:
Reviewer #3: The corrections made by the authors answer most of my
questions, notably in what regards items (1), (2), (6) (authors' "List of
edits"). Regarding the other items, I think that the changes are useful,
however, they still raise some questions which I make explicit below.
Hopefully the authors can easily answer them.
(3) I think that the example and Figure 3 (pages 12) are useful. In this
figure, it would probably be better to denote the intersection in the
RHS (skeleton) another way than the "virtual intersections" in the LHS
(the former intersections have nothing virtual as they denote actual
operations in the structure). Regarding the example shown in Figure 3,
my impression is that the numbers 2 and 5 should be exchanged in one of
the sides shown in this figure. Indeed, in the LHS, let us denote by
1' the portion of the line 1->4 between (intersection of 1 and 6) and
(intersection of 2 and 4), by 6' the portion of the line 6->3 between
(intersection of 6 and 1) and (intersection of 3 and 5), and by 5' the
portion of the line 5->2 betwen its two intersection points. Then the
LHS appears as a circuit algebra combination of (2,2) ingredients with
entries (1,6,1',6'), (5,3,6',5'), and (4,2,5',1').
In the RHS, we denote by 1',6',5' the portion of lines which prolongate
1,6,5, respectively. Then the skeleton in the RHS appears as a combination
of (2,2) ingredients (1,6,1',6'), (2,3,6',5'), and (4,5,5'1'). These
sequences of numbers agree only after one exchanges 2 and 5 in one of
the sides.
If Figure 3 is correct as shown, could the authors please communicate
a detailed proof (not to be included in the paper) of it?
(5) I think that the clarifications are useful, however I still don't see
where in the text the spaces $vT(\uparrow^n)$ and $wT(\uparrow^n)$ are
used in the sequel. It seems that the notion to be used after Definition
3.12 is rather ${\mathcal B}_n^w$.
In this situation the definitions of these spaces could be omitted.