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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Solutions:

(a) $\mathcal{S}$ is the two colinear vectors plotted in $\mathbb{R}^2$. $\mathcal{S}^\perp$ is the line passing through the origin at a 45 degree angle through the second and fourth quadrants. In other words, it makes an angle of $3\pi/4$ with the positive $x$ axis. $(\mathcal{S}^\perp)^\perp$ is the line containing the two vectors of $\mathcal{S}$, but extending infinitely in both directions.

(b) $\mathcal{S}$ is the two vectors plotted in $\mathbb{R}^2$. They are not colinear. $\mathcal{S}^\perp$ is just the zero vector. Nothing else in $\mathbb{R}^2$ is perpendicular to both vectors in $\mathcal{S}$. $(\mathcal{S}^\perp)^\perp$ is the entire plane because everything is perpendicular to the zero vector.

(c) $\mathcal{S}$ is the zero vector. $\mathcal{S}^\perp$ is the whole plane. $(\mathcal{S}^\perp)^\perp$ is the zero vector again.

(d) We need to show that it is order reversing and inflationary under inclusion. For two sets $s_1$ and $s_2$ such that $s_1\subset s_2$, $s_1^\perp$ is all the vectors orthogonal to every vector in $s_1$ and $s_2^\perp$ is all the vectors orthogonal to every vector in $s_2$. Since $s_1\subset s_2$, everything in $s_2^\perp$ is orthogonal to everything in $s_1$ because every vector in $s_1$ is a vector in $s_2$. Thus, $s_2^\perp\subset s_1^\perp$. $s_2$ may contain more vectors than $s_1$ so the vectors in $s_2^\perp$ may have to satisfy more restrictions than those in $s_1^\perp$, so the inclusion can be strict. Thus the orthogonal complement is order reversing. It is trivially inflationary because a vector is always orthogonal to a vector which is orthogonal to it, that is, $s_1\subset(s_1^\perp)^\perp$ because a vector in $s_1$ is always perpendicular to every vector in $s_1^\perp$.

(e) The closure operation is the span operation from linear algebra.

(f) It follows that the closed sets are the linear subspaces of $\mathcal{R}^n$.

(g) We showed in class that the maps restricted to the subposets defined by the image sets of each map in an abstract Galois correspondence are anti-isomorphisms. That's what's going on here: Restricting to the linear subspaces on $\mathbb{R}^n$ is restricting to the image sets because $f(g(f(s)))=f(s)$ and the closure operation is $f(g(s))$ (and both reasons reverse for the other map). Since this is true for any AGC, it is true for ours.

Endorsing all solutions above. -Steven.Jackson (talk) 14:15, 14 May 2018 (EDT)