Filters, Ideals, and Galois Connections

This post was machine-translated from the Korean original by Marvin (via Kimi). It may contain errors or awkward phrasing — the Korean original is the source of truth.

This article gathers topics scattered throughout [Bou] along with some topics not covered there, combining them into a single post.

Filter and Ideal

First, we define the following two concepts.

Definition 1 For an ordered set \(A\), a subset \(X\subseteq A\) is a lower set (resp. upper set) if whenever \(y\in A\) satisfies \(y\leq x\) (resp. \(x\leq y\)) for some \(x\in X\), then \(y\in X\).

A nonempty right directed lower set is called an ideal, and a nonempty left directed upper set is called a filter.

The set \(E\) itself is both a filter and an ideal. Filters and ideals distinct from \(E\) are called proper.

Example 2 Let an ordered set \(A\) be given. For any \(x\in A\), the downward closure¹

\[\downarrow x=\{y\in A\mid y\leq x\}\]

is an ideal of \(A\). Such an ideal is called a principal ideal.

Similarly, the upward closure

\[\uparrow x=\{y\in A\mid y\geq x\}\]

is a filter of \(A\), and such a filter is called a principal filter.

We are mostly interested in the case where \(A\) is a lattice. In this case, (§Directed Sets, ⁋Definition 4)

• A nonempty lower set \(I\) is an ideal if and only if \(x\vee y\in I\) for all \(x,y\in I\).
• A nonempty upper set \(F\) is a filter if and only if \(x\wedge y\in F\) for all \(x,y\in F\).

Example 3 Let a set \(A\) be given. Endowing \(\mathcal{P}(A)\) with the natural order relation \(\subseteq\) makes it a lattice, and in particular, for any \(X,Y\in\mathcal{P}(A)\),

\[X\vee Y=X\cup Y,\qquad X\wedge Y=X\cap Y\]

hold. In \(\mathcal{P}(A)\), the two operations \(\vee\) and \(\wedge\) additionally satisfy the distributive laws

\[X\vee(Y\wedge Z)=(X\vee Y)\wedge(X\vee Z),\qquad X\wedge(Y\vee Z)=(X\wedge Y)\vee(X\wedge Z).\]

Definition 4 Let a lattice \(A\) be given, and let \(I\) and \(F\) be a proper ideal and a proper filter of \(E\), respectively. \(I\) is a prime ideal if for any \(x,y\in A\), whenever \(x\wedge y\in I\), then either \(x\in I\) or \(y\in I\). Similarly, \(F\) is a prime filter if for any \(x,y\in A\), whenever \(x\vee y\in F\), then either \(x\in F\) or \(y\in F\). (§Field of Fractions, ⁋Proposition 8)

Equivalently, one may define \(I\) to be a prime ideal by requiring that \(A\setminus I\) be a filter.

Proposition 5 Let a lattice \(A\) be given in which the distributive law between the two operations \(\vee\) and \(\wedge\) holds. Then every maximal ideal is a prime ideal, and every maximal filter is a prime filter.

Proof

Let \(I\) be a maximal ideal, and suppose \(x\wedge y\in I\). Assume, for contradiction, that \(x,y\not\in I\). Define a new set \(J\) as the set of all \(z\) such that \(x\wedge z\in I\).

1. If \(z_1,z_2\in J\), then \(x\wedge (z_1\vee z_2)=(x\wedge z_1)\vee(x\wedge z_2)\in I\), so \(z_1\vee z_2\in J\).
2. If \(z\in J\) and \(z'\leq z\), then \(z'\in J\). Since \((x\wedge z')\vee (x\wedge z)=x\wedge (z'\vee z)=x\wedge z\), we have \(x\wedge z'\leq x\wedge z\), and since \(x\wedge z\in I\), it follows that \(x\wedge z'\in I\) as well.
3. In particular, it is clear that \(x\not\in J\) and \(y\in J\).

Therefore, \(J\) is a proper ideal strictly containing \(I\), which contradicts the maximality of \(I\). Similarly, one can show that every maximal filter is prime.

Maximal filters are also called ultrafilters.

For two ordered sets \(A,B\), consider an increasing function \(f:A\rightarrow B\). Let \(Y\) be any lower set of \(B\) and let \(X=f^{-1}(Y)\). If for some \(y\in A\) there exists \(x\in X\) such that \(y\leq x\), then \(f(y)\leq f(x)\), and since \(Y\) is a lower set, \(f(y)\in Y\), so \(y\in X\). Thus the preimage of a lower set is again a lower set.

Galois Connection

The Galois connection we are about to introduce, as its name suggests, originates from Galois theory for field extensions. However, it can be abstracted as a relation between two ordered sets, and this abstraction proves useful in many areas.

Definition 6 Let two ordered sets \(A\) and \(B\) be given.

1. Let two increasing functions \(F:A\rightarrow B\) and \(G:B\rightarrow A\) satisfy

   \[F(a)\leq b\iff a\leq G(b)\]

   for all \(a\in A\) and \(b\in B\). Then \(F\) is called the lower adjoint of \(G\), \(G\) is called the upper adjoint of \(F\), and the pair \((F,G)\) is called a monotone Galois connection between \(A\) and \(B\).

2. Let two decreasing functions \(F:A\rightarrow B\) and \(G:B\rightarrow A\) satisfy

   \[b\leq F(a)\iff a\leq G(b)\]

   for all \(a\in A\) and \(b\in B\). Then \(F\) and \(G\) are each called a polarity of the other, and the pair \((F,G)\) is called an antitone Galois connection between \(A\) and \(B\).

In either case, the function \(G\circ F:A\rightarrow A\) always satisfies \(a\leq G(F(a))\). For a monotone Galois connection,

\[a\leq G(F(a))\iff F(a)\leq F(a)\]

and the latter is always true; the same holds for an antitone Galois connection. On the other hand, the behavior of \(F\circ G\) depends on whether the connection is monotone or antitone. For a monotone Galois connection,

\[F(G(b))\leq b\iff G(b)\leq G(b)\]

and the latter is always true, so \(F(G(b))\leq b\) holds. For an antitone Galois connection,

\[b\leq F(G(b))\iff G(b)\leq G(b)\]

so \(b\leq F(G(b))\) always holds.

Meanwhile, \(G\circ F\) and \(F\circ G\) are compositions of two increasing functions or two decreasing functions, so in either case they must be increasing.

For convenience, we shall abbreviate \(G\circ F\) and \(F\circ G\) as \(GF\) and \(FG\), respectively, in the remainder of this article.

Proposition 7 Let two ordered sets \(A,B\) be given with a monotone Galois connection \(F:A\rightarrow B\) and \(G:B\rightarrow A\) between them. Then \(GFG(y)=G(y)\) holds for all \(y\in B\).

If these form an antitone Galois connection, then \(GFG(y)=G(y)\) and \(FGF(x)=F(x)\) hold for all \(x\in A\) and \(y\in B\).

Proof

First, substituting \(a=G(y)\) into \(a\leq GF(a)\) yields \(G(y)\leq GFG(y)\). On the other hand, we have shown that \(FG\) satisfies \(FG(b)\leq b\) for all \(b\in B\), and since \(G\) is increasing, we also obtain \(GFG(y)\leq G(y)\). Therefore, \(GFG(y)=G(y)\).

On the other hand, if the pair \((F,G)\) is an antitone Galois connection, \(G(y)\leq GFG(y)\) is shown in the same way as above. Also, since \(b\leq FG(b)\) always holds for all \(b\in B\), and \(G\) is decreasing, \(G(y)\geq GFG(y)\) again holds, so \(GFG(y)=G(y)\). The equality \(FGF(x)=F(x)\) follows easily by interchanging the roles of \(F\) and \(G\).

The following definition is important not only in topology but also in lattice theory.

Definition 8 For an ordered set \(A\), a function \(f:A\rightarrow A\) is a closure operator if the following three conditions are satisfied:

1. For all \(x\in A\), \(x\leq f(x)\).
2. For all \(x\in A\), \(f(x)=f(f(x))\).
3. If \(x\leq y\), then \(f(x)\leq f(y)\).

In this case, \(x\) is closed if \(f(x)=x\).

Fix an antitone Galois connection. From the result \(GFG(y)=G(y)\) in Proposition 7, substituting \(y=F(x)\) for any \(x\in A\) gives

\[GFGF(x)=GF(x)\]

Thus the function \(GF\) satisfies all the conditions above and is therefore a closure operator. Similarly, in an antitone Galois connection, \(FG\) is also a closure operator.

By definition, \(x\) and \(y\) being closed with respect to \(GF\) and \(FG\), respectively, means that \(GF(x)=x\) and \(FG(y)=y\). From Proposition 7, we know that all elements in the images of \(F\) and \(G\) are closed. Conversely, if an arbitrary element \(x\) is closed with respect to \(GF\), then \(GF(x)=x\) implies that \(x\) belongs to the image of \(G\), and the corresponding statement for \(FG\) is proved similarly.

Through this process, for a Galois connection between ordered sets \(A\) and \(B\), we can construct collections \(A'\subseteq A\) and \(B'\subseteq B\) of closed subsets, and the restrictions of \(F\) and \(G\) to these collections are well defined. Moreover, these restrictions \(F\vert_{A'}\) and \(G\vert_{B'}\) are bijections and form an anti-isomorphism. These are specifically called Galois correspondences.

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References

[Bou] N. Bourbaki, Theory of Sets. Elements of mathematics. Springer Berlin-Heidelberg, 2013. [Wikipedia] Galois connection

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1. An upper set is also called an upward closed set, and a lower set is also called a downward closed set. ↩