The relation between the left- and right- adjoints participating in Galois connections w.r.t. a common functor

Question

Consider a monotone Galois connection $(F,G_r)$. Suppose that $(G_l,F)$ is another monotone Galois connection, where $F$ is the same functor in both connections. Has the relation between $G_l$ and $G_r$ been studied? Where can I read more about it?

Answer 1

As $F(x)\leq F(x)\leq F(x)$, we have $G_l(F(x)) \leq x \leq G_r(F(x))$ for any $x$. However I doubt we can say much more, because of the two following examples:

• If $L$ has a minimal and maximal element, then $L\to \ast$ ($\ast$ is the poset with only one element) has both an upper and lower adjoint. The upper one maps the unique element of $\ast$ to the maximal element and the lower one to the minimal element. In that example, $G_l(F(x))$ is always the farthest of $G_r(F(x))$ possible. Actually we have $G_l\leq G_r$.

• If $L$ as binary meet and joins, $L\to L\times L,\,x\mapsto (x,x)$ has both a lower and upper adjoint. The lower maps $(x,y)$ to the join $x\vee y$ and the upper maps $(x,y)$ to the meet $x\wedge y$. In this example, $G_l(F(x))=G_r(F(x))$ for any $x$. But for any $x,y$, one has $x\wedge y \leq x \leq x\vee y$ so in that example $G_r\leq G_l$.

(If you try harder I'm pretty sure you can find examples when $G_l$ and $G_r$ are not comparable, globally but also even pointwise.)