Ha! Nice. I’d forgotten that they didn’t require join to be commutative. But they very clearly and intentionally do not. I don’t see any way to wriggle out of your counterexample if join isn’t commutative. (If join is commutative, then, even if a+b and b+a were distinct, they wouldn’t both be in the image of the lattice. My formulation of their theorem might still hold.)
I still have no idea whether your statement is true. It requires checking. But I hope now it is clear that no part of their proof can be trusted without some editing.
If you have enough interest to try to write a claim and a proof without references to the paper, I guess it would be nice to post it as a direct comment to the post.
Ha! Nice. I’d forgotten that they didn’t require join to be commutative. But they very clearly and intentionally do not. I don’t see any way to wriggle out of your counterexample if join isn’t commutative. (If join is commutative, then, even if a+b and b+a were distinct, they wouldn’t both be in the image of the lattice. My formulation of their theorem might still hold.)
I still have no idea whether your statement is true. It requires checking. But I hope now it is clear that no part of their proof can be trusted without some editing.
If you have enough interest to try to write a claim and a proof without references to the paper, I guess it would be nice to post it as a direct comment to the post.
btw, I mentioned the work you two have been doing here to the author and tried to get him to respond here but unfortunately he hasn’t agreed to.