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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. • Options
1.

Thanks for the great diagram!

Comment Source:Thanks for the great diagram!
• Options
2.
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=, d=, e=

$$g^* (\mathrm{d})$$=, $$g^* (\mathrm{e})$$=

$$\leq$$ and $$\nleq$$.

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=, d=, e= \$$g^* (\mathrm{d})\$$=, \$$g^* (\mathrm{e})\$$= \$$\leq\$$ and \$$\nleq\$$.