2. Polarities

We start with a context, by which is meant a triple $(X,Y,\rho)$, where $X$ and $Y$ are sets and $\rho$ is a binary relation between $X$ and $Y$, i.e., $\rho \subseteq X \times Y$.

For any subset $A$ of $X$, let $A ^ {\uparrow} = \{y \in Y: a \rho y$    for all $a \in A\}$, and for any subset $B$ of $Y$, let $B ^ {\downarrow} = \{x \in X: x \rho b$    for all $b \in B\}$. Then it can be checked that

1. $A _ 1 \subseteq A _ 2 \Rightarrow A _ 1 ^ {\uparrow} \supseteq A _ 2 ^ {\uparrow}$ and $B _ 1 \subseteq B _ 2 \Rightarrow B _ 1 ^ {\downarrow} \supseteq B _ 2 ^ {\downarrow}$;

2. $A \subseteq A ^ {\uparrow \downarrow}$ and $B \subseteq B ^ {\downarrow \uparrow}$ (where $A ^ {\uparrow \downarrow}$ means $(A ^ {\uparrow}) ^ {\downarrow}$, etc.);

3. $A ^ {\uparrow \downarrow \uparrow} = A ^ {\uparrow}$ and $B ^ {\downarrow \uparrow \downarrow} = B ^ {\downarrow}$;

4. $A \mapsto \bar A$ and $B \mapsto \hat B$ are closure operations, where $\bar A = A ^ {\uparrow \downarrow}$ and $\hat B = B ^ {\downarrow \uparrow}$;

5. the lattice of closed subsets of $X$ is dually isomorphic to the lattice of closed subsets of $Y$.

The maps $A \mapsto A ^ {\uparrow}$ and $B \mapsto B ^ {\downarrow}$ are said to constitute a polarity, and the correspondence of closed subsets is called a Galois correspondence, after the example of Galois theory mentioned below.

Examples.

(1) The completion by cuts CC$(P)$: $(P,P, \leq )$

(2) Orthogonality: $x \rho y$ is $x \perp y$ in one sense or another

(a) $X = Y =$   R$^ n$

(b) Like (a) but for Hilbert space.

(c) $X = Y = R$, a ring; $x \rho y$ means $xy = 0$.

(3) Commuting: $x \rho y$ is $yx = xy$

(a) $X = Y = G$, a group

(b) $X = Y = R$, a ring

(4) Vanishing: $Y$ consists of functions on $X$ to some group; $x \rho f$ means $f(x) = 0$.

(a) $X$ is any subset of the reals R, $Y$ is the set of continuous functions $f: X \rightarrow$   R;

(b) $X = V$, a vector space over a field $F$; $Y = V ^ *$, the ``dual space'' consisting of all linear functionals on $V$, i.e., all linear maps $f: V \rightarrow$   F;

(c) $X = B$, a Banach space over C; $Y = B ^ *$, the ``dual space'' consisting of all bounded linear functionals $f: B \rightarrow$   C (the complex numbers);

(d) $X =$   C$^ n$, $Y =$   C$[X _ 1,\dots, X _ n]$, the ring of polynomials in $n$ variables $X _ 1,\dots, X _ n$;

(e) $X = G$, a finite group, $Y$ is the group of characters of $X$, i.e., homomorphisms $\chi$ of $G$ into the unit-circle group; here $x \rho \chi$ means $\chi (x) = 1$.

(5) Fixing: $X$ is some set, $Y$ is a set of functions from $X$ to $X$, and $x \rho f$ means $f(x) = x$.

(a) $X = K$, the smallest subfield of C containing all roots of some polynomial; $Y =$   Aut$(K)$, the group of automorphisms of $K$;

(b) $X$ is a set, and $Y = G$, a group acting on $X$; $x \rho g$ means $gx = x$.

(c) The specific case of (b) where the group $G$ acts on itself by conjugation: $\lambda _ g (x) = g ^ {-1}x g$. (Is this equivalent to another example mentioned?)

(6) Convexity:

(a) $X =$   R$^ n$, $Y$ is the space of affine functionals on R$^ n$, and x$\rho f$ means $f($x$) > 0$.

(An affine functional is a map $f:$   R$^ n \rightarrow$   R of the form $f($x$) =$   x$\cdot$   a$+$   b, for constant vectors a$,$   b$\in$   R$^ n$. In other words, x$\mapsto f($x$)-f($0$)$ is a linear transformation.)

(b) $X =$   R$^ {n+1}$, $Y$ is the dual space of linear functionals on R$^ {n+1}$, and x$\rho f$ means $f($x$) > 0$.

(b') $X = Y =$   R$^ {n+1}$ and x$\rho$   y means x$\cdot$   y$> 0$.

(7) Hull-kernel constructions of closure systems; $\rho$ is $\in$:

(a) $X =$   R$^ n$ and $Y$ is the set of closed half-spaces (= 6(a));

(b) $X = L$, a distributive lattice, and $Y = \Pi _ L$, the set of prime ideals of $L$;

(c) $X = R$, a commutative ring with 1, and $Y$ is the set of prime ideals of $R$;

(d) $X = R$, a commutative ring with 1, and $Y$ is the set of maximal ideals of $R$.

(8) Model theory: $\rho$ is satisfaction.

(a) First-order model theory, say for groups: $X$ is the class of all groups, $Y$ is the set of all first-order sentences in the language of groups, and $G \rho S$ means that $G$ satisfies the sentence $S$, i.e., $S$ is true about $G$. (Here $X$ is a class instead of a set, but polarities work the same as usual.)

(b) Equational model theory, say for lattices: $X$ is the class of all lattices, $Y$ is the set of all possible laws involving $\wedge$ and $\vee$, and $\rho$ is satisfaction.

(9) Concept lattices (R. Wille and his co-workers)

$X$ is a finite set of objects, $Y$ is a finite set of attributes, and $x \rho y$ means that the object $x$ has the attribute $y$. A concept is a pair $(A,B)$ with $A$ a closed subset of $X$, $B$ a closed subset of $Y$, and $A \leftrightarrow B$ under the Galois correspondence. We write $(A _ 1, B _ 1) \leq (A _ 2, B _ 2)$ if $A _ 1 \subseteq A _ 2$.

See the last page of this handout for some examples.