Math 380, Spring 2018, Assignment 13

"Reeling and Writhing, of course, to begin with," the Mock Turtle replied; "And then the different branches of Arithmetic - Ambition, Distraction, Uglification, and Derision."

- Lewis Carroll, Alice's Adventures in Wonderland

Read:

1. Notes on abstract Galois correspondences.

Carefully define the following terms, then give one example and one non-example of each:

1. Poset.
2. Homomorphism (of posets; a.k.a. order-preserving map).
3. Isomorphism (of posets).
4. Anti-homomorphism (of posets; a.k.a. order-reversing map).
5. Anti-isomorphism (of posets).
6. Inflationary pair (of maps betweeen posets).
7. Abstract Galois correspondence.
8. Closure (of an element of a poset occurring in an abstract Galois correspondence).
9. Closed element (of a poset occurring in an abstract Galois correspondence).
10. Radical ideal.

Carefully state the following theorems (you do not need to prove them):

1. Theorem relating three passes through an abstract Galois correspondence to one pass.
2. Theorem concerning the restriction of an abstract Galois correspondence to the actual images of its mappings.
3. Theorem characterizing the closure of an element ("The closure $\overline{s}$ of $s$ is the smallest...")
4. Theorem characterizing the smallest variety containing a given set $A\subseteq\mathsf{k}^n$ (a.k.a. the Zariski closure of $A$).
5. Theorem relating the output of $\mathbb{I}$ to radical ideals.

Solve the following problems:

1. (Orthogonal complement as abstract Galois correspondence) Working in $\mathbb{R}^n$ with its usual dot product, define the orthogonal complement of a set $S$ to be the set of all vectors in $\mathbb{R}^n$ that are orthogonal to all elements of $S$, i.e. $$S^{\perp}=\{\vec{x}\in\mathbb{R}^n\,|\,\vec{x}\cdot\vec{s}=0\quad\forall\vec{s}\in S\}.$$

(a) Take $n=2$ and $S=\{(1,1),(2,2)\}$. Draw pictures of $S$, of $S^{\perp}$, and of the double complement $\left(S^\perp\right)^\perp$.

(b) Take $n=2$ and $S=\{(1,1),(2,0)\}$. Draw pictures of $S$, of $S^{\perp}$, and of the double complement $\left(S^\perp\right)^\perp$.

(c) Take $n=2$ and $S=\{(0,0)\}$. Draw pictures of $S$, of $S^{\perp}$, and of the double complement $\left(S^\perp\right)^\perp$.

(d) Now let both $\mathcal{S}$ and $\mathcal{T}$ denote the power set of $\mathbb{R}^n$, regarded as a poset under inclusion. Define $f:\mathcal{S}\rightarrow\mathcal{T}$ by $f(S)=S^\perp$ and define $g:\mathcal{T}\rightarrow\mathcal{S}$ by $g(T)=T^\perp$. (Of course $f$ and $g$ are really the same object with two different names, as are $\mathcal{S}$ and $\mathcal{T}$. We are about to set up a Galois correspondence from a certain poset to itself.) Show that the pair $(f,g)$ is an abstract Galois correspondence between $\mathcal{S}$ and $\mathcal{T}$.

(e) Describe the closure operation in the Galois correspondence defined above. (Hint: you have already met this operation in elementary Linear Algebra).

(f) Describe the closed elements in the Galois correspondence defined above.

(g) Show that the set of linear subspaces of $\mathbb{R}^n$, regarded as a poset under inclusion, is anti-isomorphic to itself. Describe the anti-isomorphism explicitly, with pictures for small $n$.

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