Exercise 90 - Chapter 1

Fredrick Eisele

Galois connection a.k.a. Adjunction Formula $$ f(p) \le q \iff p \le g(q) \\ \text{Where g is right-adjoint to f and f is left-adjoint to g} $$

1. Choose a nontrivial partition \( c : S \twoheadrightarrow P \) and let \( g_!(c) \) be its push forward partition on T.
2. Choose any coarser partition \( d : T \twoheadrightarrow P' \), i.e. where \( g_!(c) \le d \) .
3. Choose any non-coarser partition \( e : T \twoheadrightarrow Q \), i.e. where \( g_!(c) \nleq e \). (If you can’t do this, revise your answer for #1.)
4. Find \( g^*(d) \text{ and } g^*(e) \) .
5. The adjunction formula, in this case, says that since \( g_!(c) \le d \text{ and } g_!(c) \nleq e \) , we should have \( c \le g^*(d) \text{ and } c \nleq g^*(e) \) . Show that this is true.

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Ignacio Viglizzo

Thanks for the great diagram!

David Lambert

The easiest, and probably most illuminating, method is to look at the diagram above. Nevertheless, I thought I would give an explicit example.

\(S=\{1,2,3,4\},\,\,T=\{12,3,4\}\).

c=[124][3], d=[1234], e=[12][3][4]

\(g^* (\mathrm{d})\)=[1234], \(g^* (\mathrm{e})\)=[12][3][4]

[124][3]\(\leq\)[1234] and [124][3]\(\nleq\)[12][3][4].