Thank you @ValterSorana.

I think my 'bug' was, that I did understand how x, y (or a, b) are selected to work within

$$ f(a) \le_B b \textrm{ if and only if } a \le_A g(b) $$

For some reason, I thought that f(a) = b . Which is why, it never entered my mind, to try b to be some other arbitrary number...

But when I re-read John's definition at the beginning of this lecture, there was a definition of a monotone map, right above the definition of the Galois connection.

And in that definition, there was:

for *all* elements \\(x,y \in A\\),

So my two functions needed to work for *any* x, y -- which is clearly not the case with my g and f.

But then, it seems like there are not a lot of pairs of functions that can do the above.

At least, I am trying to develop, as per John's ask, some intuition -- of how to find those functions.

And all I can come up with so far is: find monotone functions that have inverses. Then see if the inverse part of the relationship can be relaxed, while retaining the need Galois connection properties.

So f(x) = x*2 and f(x)=x/2 seem to work. However this, probably, too trivial to be useful. So I am not sure if I am on the right track in developing the needed intuition.

I also found this (by searching, no intuition) that inverse operation of strictly diagonally dominant matrix is [monotone](https://math.stackexchange.com/questions/972725/show-that-the-inverse-of-a-strictly-diagonally-dominant-matrix-is-monotone?rq=1)

But not sure where to even start to see if that operation forms a galois connection.

I think my 'bug' was, that I did understand how x, y (or a, b) are selected to work within

$$ f(a) \le_B b \textrm{ if and only if } a \le_A g(b) $$

For some reason, I thought that f(a) = b . Which is why, it never entered my mind, to try b to be some other arbitrary number...

But when I re-read John's definition at the beginning of this lecture, there was a definition of a monotone map, right above the definition of the Galois connection.

And in that definition, there was:

for *all* elements \\(x,y \in A\\),

So my two functions needed to work for *any* x, y -- which is clearly not the case with my g and f.

But then, it seems like there are not a lot of pairs of functions that can do the above.

At least, I am trying to develop, as per John's ask, some intuition -- of how to find those functions.

And all I can come up with so far is: find monotone functions that have inverses. Then see if the inverse part of the relationship can be relaxed, while retaining the need Galois connection properties.

So f(x) = x*2 and f(x)=x/2 seem to work. However this, probably, too trivial to be useful. So I am not sure if I am on the right track in developing the needed intuition.

I also found this (by searching, no intuition) that inverse operation of strictly diagonally dominant matrix is [monotone](https://math.stackexchange.com/questions/972725/show-that-the-inverse-of-a-strictly-diagonally-dominant-matrix-is-monotone?rq=1)

But not sure where to even start to see if that operation forms a galois connection.