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# Exercise 90 - Chapter 1

edited June 2018

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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Thanks for the great diagram!

Comment Source:Thanks for the great diagram!
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edited July 2018

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].

Comment Source: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].