diff --git a/iii/alggeom/03_schemes.tex b/iii/alggeom/03_schemes.tex index 9f62ef0..913f6a0 100644 --- a/iii/alggeom/03_schemes.tex +++ b/iii/alggeom/03_schemes.tex @@ -149,10 +149,15 @@ \subsection{Definitions and examples} \[ \mathcal O_U(U) = k[x,x^{-1},y] \cap k[x,y,y^{-1}] \subseteq k[x,x^{-1},y,y^{-1}] \] Thus, \( \mathcal O_U(U) = k[x,y] \). This is a contradiction: one way to see this is that there exists a maximal ideal \( (x, y) \) in the ring of global sections in \( (U, \mathcal O_U) \) with empty vanishing locus. + + In general, if \( X \) is a scheme, \( f \in \Gamma(X, \mathcal O_X) = \mathcal O_X(X) \), and \( p \in X \), then there is a well-defined stalk \( \mathcal O_{X,p} \) at \( p \), which is of the form \( A_{\mathfrak p} \) up to isomorphism, where \( \mathfrak p \) is a prime ideal. + To say this, we are using an isomorphism of an open set \( V_p \) containing \( p \) to \( \Spec A \). + In particular, \( A_{\mathfrak p} \) has a unique maximal ideal, namely \( \mathfrak p A_{\mathfrak p} \). + We say that \( f \) vanishes at \( p \) if its image in \( \faktor{A_{\mathfrak p}}{\mathfrak p A_{\mathfrak p}} \), or equivalently, \( f \in \mathfrak p A_{\mathfrak p} \). + As a consequence, the vanishing locus \( \mathbb V(f) \subseteq X \) is well-defined. \end{example} \subsection{Gluing sheaves} -% TODO: move? Let \( X \) be a topological space with a cover \( \qty{U_\alpha} \). Let \( \qty{\mathcal F_\alpha} \) be sheaves on \( \qty{U_\alpha} \), with isomorphisms \[ \varphi_{\alpha\beta} : \eval{\mathcal F_\alpha}_{U_\alpha \cap U_\beta} \to \eval{\mathcal F_\beta}_{U_\alpha \cap U_\beta} \] @@ -181,10 +186,78 @@ \subsection{Gluing sheaves} \[ \varphi_{\alpha\beta} \circ \varphi_{\gamma\alpha}\qty(\eval{s}_{V \cap U_\alpha \cap U_\beta}) = \varphi_{\gamma\beta}\qty(\eval{s}_{V \cap U_\alpha \cap U_\beta}) \] \end{proof} -\subsection{???} +\subsection{Gluing schemes} +Let \( (X, \mathcal O_X) \) and \( (Y, \mathcal O_Y) \) be schemes with open sets \( U \subseteq X, V \subseteq Y \), and let \( \varphi : (U, \eval{\mathcal O_X}_U) \to (V, \eval{\mathcal O_Y}_V) \) be an isomorphism. +The topological spaces \( X, Y \) can be glued on \( U, V \) using \( \varphi \). + +First, take \( S = \faktor{X \sqcup Y}{U \sim V} \). +By definition of the quotient topology, the images of \( X \) and \( Y \) in \( S \) form an open cover, and their intersection is the image of \( U \), or equivalently, the image of \( V \). +Now, we can glue the structure sheaves on these open sets as described in the previous subsection. +Note that in this case, there is no cocycle condition. +\begin{example}[the bug-eyed line; the line with doubled origin] + Let \( k \) be a field. + Let \( X = \Spec k[t] \) and \( Y = \Spec k[u] \). + Let + \[ U = \Spec k[t, t^{-1}] = \Spec k[t]_t = U_t \subseteq X;\quad V = \Spec k[u, u^{-1}] = \Spec k[u]_u = U_u \subseteq Y \] + We define the isomorphism \( \varphi : U \to V \) given by \( t \mapsfrom u \). + Technically, we define an isomorphism of rings \( k[u, u^{-1}] \to k[t, t^{-1}] \) by \( u \mapsto t \) and then apply \( \Spec \). + At the level of topological spaces, \( X = \mathbb A^1_k \) and \( Y = \mathbb A^1_k \), so \( U = \mathbb A^1_k \setminus \qty{(t)} \) and \( V = \mathbb A^1_k \setminus \qty{(u)} \). + Gluing along this isomorphism, we obtain a scheme \( S \) which is a copy of \( \mathbb A^1_k \) but with two origins. + Note that the generic points in \( X \) and \( Y \) lie in \( U \) and \( V \) respectively, and thus are glued into a single generic point in \( S \). + + Consider the open sets in \( S \). + Open sets entirely contained within \( X \) and \( Y \) yield open sets in \( S \). + We also have open sets of the form \( W = S \setminus \qty{\mathfrak p_1, \dots, \mathfrak p_r} \) where \( \mathfrak p_i \) is contained in \( U \) or \( V \). + One example is \( W = S \); we can calculate \( \mathcal O_S(S) \) using the sheaf axioms, and one can show that it is isomorphic to \( k[t] \). + We can conclude that \( S \) is not an affine scheme, because there is a maximal ideal in \( k[t] \) where the vanishing locus is precisely two points. +\end{example} +\begin{example}[the projective line] + Let \( X = \Spec k[t] \) and \( Y = \Spec k[s] \), and define \( U = \Spec k[t,t^{-1}], V = \Spec k[s,s^{-1}] \) as above. + We glue these schemes using the isomorphism \( s \mapsto t^{-1} \), giving the projective line \( \mathbb P^1_k \). +\end{example} +\begin{proposition} + \( \mathcal O_{\mathbb P^1_k}(\mathbb P^1_k) = k \). +\end{proposition} +% this does not require that k is algebraically closed +\begin{proof}[Proof sketch] + We use the same idea as in the previous example. + The only elements of \( k[t, t^{-1}] \) that are both polynomials in \( t \) and \( t^{-1} \) are the constants. + % important exercise. +\end{proof} +In particular, \( \mathbb P^1_k \) is not an affine scheme. +\begin{example} + We can similarly build a scheme \( S \) which is a copy of \( \mathbb A^2_k \) with a doubled origin. + This has the interesting property that there exist affine open subschemes \( U_1, U_2 \subseteq S \) such that \( U_1 \cap U_2 \) is not affine; we can take \( U_1 \) and \( U_2 \) to be \( S \) but with one of the origins deleted. + Note that \( \mathbb A^1_k \) without the origin is affine. +\end{example} + +Let \( \qty{X_i}_{i \in I} \) be schemes, \( X_{ij} \subseteq X_i \) be open subschemes, and \( f_{ij} : X_{ij} \to X_{ji} \) be isomorphisms such that +\[ f_{ii} = \id_{X_i};\quad f_{ij} = f_{ji}^{-1};\quad f_{ik} = f_{jk} \circ f_{ij} \] +where the last equality holds whenever it is defined. +Then there is a unique scheme \( X \) with an open cover by the \( X_i \), glued along these isomorphisms. +This is an elaboration of the above construction, which is discussed on the first example sheet. + +Let \( A \) be a ring, and let \( X_i = \Spec A\qty[\frac{x_0}{x_i}, \dots, \frac{x_n}{x_i}] \). +Let \( X_{ij} = \mathbb V\qty(\frac{x_j}{x_i})^c \subseteq X_i \). +We define the isomorphisms \( X_{ij} \to X_{ji} \) by \( \frac{x_k}{x_i} \mapsto \frac{x_k}{x_j} \qty(\frac{x_i}{x_j})^{-1} \). +The resulting glued scheme is called \emph{projective \( n \)-space}, denoted \( \mathbb P^n_A \). +% exercise: \mathcal O_{\mathbb P^n_A}(\mathbb P^n_A) = A. + +\subsection{The Proj construction} +% Idea: +% Spec : Rings -> Schemes; Proj : GradedRings -> Schemes +\begin{definition} + A \emph{\( \mathbb Z \)-grading} on a ring \( A \) is a decomposition + \[ A = \bigoplus_{i \in \mathbb Z} A_i \] + as abelian groups, such that \( A_i A_j \subseteq A_{i+j} \). +\end{definition} +\begin{example} + Let \( A = k[x_0, \dots, x_n] \), and let \( A_d \) be the set of degree \( d \) homogeneous polynomials, together with the zero polynomial. +\end{example} \begin{example} - Let \( (X, \mathcal O_X) \) and \( (Y, \mathcal O_Y) \) be schemes with open sets \( U \subseteq X, V \subseteq Y \), and let \( \varphi : (U, \mathcal \eval{O_X}_U) \to (V, \mathcal \eval{O_Y}_V) \) be an isomorphism. - The topological spaces \( X, Y \) can be glued on \( U, V \) using \( \varphi \). - We can similarly glue the relevant sheaves together, thus gluing \( X \) and \( Y \) together as schemes. - Note that in this case, there is no cocycle condition. + Let \( I \subseteq k[x_0, \dots, x_n] \) be a homogeneous ideal; that is, an ideal generated by homogeneous elements of possibly different degrees. + Then, for \( A = k[x_0, \dots, x_n] \), the ring \( \faktor{A}{I} \) is also naturally graded. + % how? \end{example} +Note that by definition, \( A_0 \) is a subring of \( A \). +We will always assume that the degree 1 elements of a graded ring generate \( A \) as an algebra over \( A_0 \). diff --git a/iii/cat/04_limits.tex b/iii/cat/04_limits.tex index dc68de6..49ea4cc 100644 --- a/iii/cat/04_limits.tex +++ b/iii/cat/04_limits.tex @@ -154,5 +154,100 @@ \subsection{Limits} Thus a cone is a span that completes the commutative square. A limit for the cospan is the universal way to complete this commutative square, which is called a \emph{pullback} of \( f \) and \( g \). Dually, colimits of spans are called \emph{pushouts}. + + If any category \( \mathcal C \) has binary products and equalisers, we can construct all pullbacks. + First, we construct the product \( A \times B \), then we form the equaliser of \( f \pi_1, g \pi_2 : A \times B \rightrightarrows C \). + This yields the pullback. + \item Let \( M \) be the two-element monoid \( \qty{1, e} \) with \( e^2 = e \). + A diagram of shape \( M \) in a category \( \mathcal C \) is an object of \( \mathcal C \) equipped with an idempotent endomorphism. + A cone over this diagram is a morphism \( f : B \to A \) such that \( ef = f \). + A limit (respectively colimit) is the monic (respectively epic) part of a splitting of \( e \). + This is because the pair \( (e, 1_A) \) has an equaliser if and only if \( e \) splits. + % Note that the functor \( F : \mathbf{Set} \to [M, \mathbf{Set}] \) is the constant map \( \Delta \), which explains why the left and right adjoints coincide. % what is 3.2(e)? + \item Let \( \mathbb N \) be the poset category of the natural numbers. + A diagram of shape \( \mathbb N \) is a \emph{direct sequence} of objects, which consists of objects \( A_0, A_1, \dots \) and morphisms \( f_i : A_i \to A_{i+1} \). + A colimit for this diagram is a \emph{direct limit}, which consists of an object \( A_\infty \) and morphisms \( g_i : A_i \to A_\infty \) which are compatible with the \( f_i \). + Dually, an \emph{inverse sequence} is a diagram of shape \( \mathbb N^\cop \), and a limit for this diagram is called an \emph{inverse limit}. + For example, an infinite-dimensional CW-complex \( X \) is the direct limit of its \( n \)-dimensional skeletons in \( \mathbf{Top} \). + The ring of \( p \)-adic integers is the limit of the inverse sequence defined by \( A_n = \faktor{\mathbb Z}{p^n\mathbb Z} \) in \( \mathbf{Rng} \). + \end{enumerate} +\end{example} +\begin{lemma} + Let \( \mathcal C \) be a category. + \begin{enumerate} + \item If \( \mathcal C \) has equalisers and all small products, then \( \mathcal C \) has all small limits. + \item If \( \mathcal C \) has equalisers and all finite products, then \( \mathcal C \) has all finite limits. + \item If \( \mathcal C \) has pullbacks and a terminal object, then \( \mathcal C \) has all finite limits. + \end{enumerate} +\end{lemma} +Note that the empty product is implicitly included in (i) and (ii). +A terminal object is a product over no factors. +\begin{proof} + \emph{Parts (i) and (ii).} + We prove (i) and (ii) in the same way. + Let \( D : J \to \mathcal C \) be a diagram. + We form the products + \[ P = \prod_{j \in \ob J} D(j);\quad Q = \prod_{\alpha \in \mor J} D(\cod \alpha) \] + These are small or finite as required. + We have morphisms \( f, g : P \rightrightarrows Q \) defined by + \[ \pi_\alpha f = \pi_{\cod \alpha};\quad \pi_\alpha g = D(\alpha) \pi_{\dom \alpha} \] + Let \( e : E \to P \) be an equaliser for \( f \) and \( g \), and define \( \lambda_j = \pi_j e : E \to D(j) \). + These morphisms form a cone, since if \( \alpha : j \to j' \), we have + \[ D(\alpha) \lambda_j = D(\alpha) \pi_j e = \pi_\alpha g e = \pi_\alpha f e = \pi_{j'} e = \lambda_{j'} \] + Given any cone \( (A, (\mu_j)_{j \in \ob J}) \) over \( D \), we have a unique \( \mu : A \to P \) with \( \pi_j \mu = \mu_j \) for all \( j \). + Then, + \[ \pi_\alpha f \mu = \mu_{\cod \alpha} = D(\alpha) \mu_{\dom \alpha} = \pi_\alpha g \mu \] + for all \( \alpha \), so \( \mu \) factors uniquely through \( e \). + + \emph{Part (iii).} + We show that the hypotheses of (iii) imply those of (ii). + If \( 1 \) is the terminal object, we form the pullback of the span + % https://q.uiver.app/#q=WzAsMyxbMSwwLCJBIl0sWzEsMSwiMSJdLFswLDEsIkIiXSxbMCwxXSxbMiwxXV0= +\[\begin{tikzcd} + & A \\ + B & 1 + \arrow[from=1-2, to=2-2] + \arrow[from=2-1, to=2-2] +\end{tikzcd}\] + This has the universal property of the product \( A \times B \), so \( \mathcal C \) has binary products and hence all finite products by induction. + To construct the equaliser of \( f, g : A \rightrightarrows B \), we consider the pullback of + % https://q.uiver.app/#q=WzAsMyxbMSwwLCJBIl0sWzEsMSwiQSBcXHRpbWVzIEIiXSxbMCwxLCJBIl0sWzAsMSwiKDFfQSwgZikiXSxbMiwxLCIoMV9BLCBnKSIsMl1d +\[\begin{tikzcd} + & A \\ + A & {A \times B} + \arrow["{(1_A, f)}", from=1-2, to=2-2] + \arrow["{(1_A, g)}"', from=2-1, to=2-2] +\end{tikzcd}\] + Any cone over this diagram has its two legs \( C \rightrightarrows A \) equal, so a pullback is an equaliser for \( f, g \). +\end{proof} +\begin{definition} + A category is called \emph{complete} if it has all small limits, and \emph{cocomplete} if it has all small colimits. +\end{definition} +\begin{example} + The categories \( \mathbf{Set}, \mathbf{Gp}, \mathbf{Top} \) are complete and cocomplete. +\end{example} + +\subsection{Preservation and creation} +\begin{definition} + Let \( G : \mathcal D \to \mathcal C \) be a functor. + We say that \( G \) + \begin{enumerate} + \item \emph{preserves} limits of shape \( J \) if whenever \( D : J \to \mathcal D \) is a diagram with limit cone \( (L, (\lambda_j)_{j \in \ob J}) \), the cone \( (GL, (G\lambda_j)_{j \in \ob J}) \) is a limit for \( GD \); + \item \emph{reflects} limits of shape \( J \) if whenever \( D : J \to \mathcal D \) is a diagram and \( (L, (\lambda_j)_{j \in \ob J}) \) is a cone such that \( (GL, (G\lambda_j)_{j \in \ob J}) \) is a limit for \( GD \), then \( (L, (\lambda_j)_{j \in \ob J}) \) is a limit for \( D \); + \item \emph{creates} limits of shape \( J \) if whenever \( D : J \to \mathcal D \) is a diagram with limit cone \( (M, (\mu_j)_{j \in \ob J}) \) for \( GD \) in \( \mathcal C \), there exists a cone \( (L, (\lambda_j)_{j \in \ob J}) \) over \( D \) such that \( (GL, (G\lambda_j)_{j \in \ob J}) \cong (M, (\mu_j)_{j \in \ob J}) \) in \( \operatorname{Cone}(GD) \), and any such cone is a limit for \( D \). + \end{enumerate} +\end{definition} +We typically assume in (i) that \( \mathcal D \) has all limits of shape \( J \), and we assume in (ii) and (iii) that \( \mathcal C \) has all limits of shape \( J \). +With these assumptions, \( G \) creates limits of shape \( J \) if and only if \( G \) preserves and reflects limits, and \( \mathcal D \) has all limits of shape \( J \). +\begin{corollary} + In any of the statements of the previous lemma, we can replace both instances of `\( \mathcal C \) has' by either `\( \mathcal D \) has and \( G : \mathcal D \to \mathcal C \) preserves' or `\( \mathcal C \) has and \( G : \mathcal D \to \mathcal C \) creates'. +\end{corollary} +\begin{example} + \begin{enumerate} + \item The forgetful functor \( U : \mathbf{Gp} \to \mathbf{Set} \) creates all small limits. + It does not preserve colimits, as in particular it does not preserve coproducts. + \item The forgetful functor \( U : \mathbf{Top} \to \mathbf{Set} \) preserves all small limits and colimits, but does not reflect them, as we can retopologise the apex of a limit cone. + \item The inclusion \( \mathbf{AbGp} \to \mathbf{Gp} \) reflects coproducts, but does not preserve them. + A free product of two groups \( G, H \) is always nonabelian, except for the case where either \( G \) or \( H \) is the trivial group, but the coproduct of the trivial group with \( H \) is isomorphic to \( H \) in both categories. \end{enumerate} \end{example} diff --git a/iii/mtncl/02_quantifier_elimination.tex b/iii/mtncl/02_quantifier_elimination.tex index d08a9e4..633c7c2 100644 --- a/iii/mtncl/02_quantifier_elimination.tex +++ b/iii/mtncl/02_quantifier_elimination.tex @@ -531,5 +531,34 @@ \subsection{Interaction with other properties} The field of fractions of \( \mathcal C \) can be made an ordered field in a canonical way, by saying \( \frac{a}{b} > 0 \) if \( ab > 0 \). The embedding \( \mathcal C \) into \( \mathcal A \) is an injective homomorphism of ordered rings, into an ordered field. By the universal property of the fraction field, there is a unique homomorphism of ordered fields from \( FF(\mathcal C) \) to \( \mathcal A \) that extends the inclusion of \( \mathcal C \) into \( \mathcal A \). - Let \( \mathcal A' \) be the real closure of \( FF(\mathcal C) \). + Let \( \mathcal A' \) be the real closure of \( FF(\mathcal C) \), so that \( \mathcal C \subseteq FF(\mathcal C) \subseteq \mathcal A' \subseteq \mathcal A \). + If \( \mathcal B \vDash \mathsf{RCF} \) and \( \mathcal C \subseteq B \), then by the same argument we have a unique ordered ring homomorphism \( FF(\mathcal C) \to \mathcal B \) extending the embedding \( \mathcal C \subseteq \mathcal B \). + Thus \( \mathcal A' \subseteq \mathcal B \) as well, and this embedding fixes \( \mathcal C \). \end{example} +\begin{corollary}[Hilbert's Nullstellensatz] + Let \( k \) be an algebraically closed field, and \( I \) be a proper ideal of \( k[x_1, \dots, x_n] \). + Then there exists \( \vb a \in k^n \) such that \( f(\vb a) = 0 \) for all \( I \in f \). +\end{corollary} +\begin{proof} + By Zorn's lemma, every proper ideal can be extended to a maximal ideal, so without loss of generality we may assume that \( I \) is a maximal ideal. + Let \( L \) be the residue field \( \faktor{k[x_1, \dots, x_n]}{I} \), and let \( \overline L \) be its algebraic closure. + By Hilbert's basis theorem, there exists a finite set of generators \( f_1, \dots, f_r \) for \( I \). + Note that \( \vb 0 \) is a witness to + \[ \overline L \vDash \exists \vb x.\, \qty(f_1(\vb x) = 0 \wedge \dots \wedge f_r(\vb x) = 0) \] + We have embeddings \( k \subseteq L \subseteq \overline L \), where both \( k \) and \( \overline L \) are algebraically closed fields. + The theory of algebraically closed fields has quantifier elimination, so is model-complete. + Thus the embedding \( k \subseteq \overline L \) is elementary, so + \[ k \vDash \exists \vb x.\, \qty(f_1(\vb x) = 0 \wedge \dots \wedge f_r(\vb x) = 0) \] + We can then take \( \vb a \) to be a witness to this existential. +\end{proof} +\begin{corollary}[Chevalley's theorem] + Let \( k \) be an algebraically closed field. + Then the image of a constructible set in \( k^n \) under a polynomial map is constructible. +\end{corollary} +\begin{proof} + The quantifier-free-definable subsets of \( k^n \) are precisely the finite Boolean combinations of the Zariski closed subsets of \( k^n \), which are by definition the constructible sets. + As \( \mathsf{ACF} \) has quantifier elimination, these are exactly the definable subsets using arbitrary formulae. + Now, if \( X \subseteq k^n \) is constructible and \( p : k^n \to k^m \) is a polynomial map, then + \[ p(X) = \qty{y \in k^m \mid \exists x.\, p(x) = y} \] + This is definable in the same language, so is a constructible set. +\end{proof} diff --git a/iii/mtncl/03_ultraproducts.tex b/iii/mtncl/03_ultraproducts.tex index e69de29..7cae99a 100644 --- a/iii/mtncl/03_ultraproducts.tex +++ b/iii/mtncl/03_ultraproducts.tex @@ -0,0 +1,65 @@ +\subsection{Products} +We will use the symbol \( \lambda \) to define functions without giving them explicit names. +The syntax \( \lambda x.\, y \) represents the function \( f \) such that \( f(x) = y \). + +Let \( \qty{\mathcal M_i}_{i \in I} \) be a set of \( \mathcal L \)-structures. +The \emph{product} \( \prod_{i \in I} \mathcal M_i \) of this family is the \( \mathcal L \)-structure with carrier set +\[ \prod_{i \in I} \mathcal M_i = \qty{\alpha : I \to \bigcup M_i \midd \alpha(i) \in \mathcal M_i} \] +such that +\begin{itemize} + \item an \( n \)-ary function symbol \( f \) is interpreted as + \[ f^{\prod_I \mathcal M_i} : \qty(\prod_I \mathcal M_i)^n \to \prod_I \mathcal M_i \] + given by + \[ (\alpha_1, \dots, \alpha_n) \mapsto \lambda i.\, f^{\mathcal M_i}(\alpha_1(i), \dots, \alpha_n(i)) \] + \item an \( n \)-ary relation symbol \( R \) is interpreted as the subset + \[ R^{\prod_I \mathcal M_i} \subseteq \qty(\prod_I \mathcal M_i)^n \] + given by + \[ R^{\prod_I \mathcal M_i} = \qty{(\alpha_1, \dots, \alpha_n) \in \qty(\prod_I \mathcal M_i)^n \midd \forall i \in I.\, (\alpha_1(i), \dots, \alpha_n(i)) \in R^{\mathcal M_i}} \] +\end{itemize} +The relation symbols in this kind of product are not particularly useful. +We want to construct a different kind of product in such a way that \( \varphi \) holds in the product if the set of \( \mathcal M_i \) that model \( \varphi \) is `large'. + +\subsection{Lattices} +\begin{definition} + A \emph{lattice} is a set \( L \) equipped with binary operations \( \wedge \) and \( \vee \) that are associative and commutative, and satisfy the \emph{absorption laws} + \[ a \vee (a \wedge b) = a;\quad a \wedge (a \vee b) = a \] + A lattice is called + \begin{itemize} + \item \emph{distributive}, if \( a \wedge (b \vee c) = (a \wedge b) \vee (a \wedge c) \); + \item \emph{bounded}, if there are elements \( \bot \) and \( \top \) such that \( a \vee \bot = a \) and \( a \wedge \top = a \); + \item \emph{complemented}, if it is bounded and for each \( a \in L \) there exists \( a^\star \in L \) called its \emph{complement} such that \( a \wedge a^\star = \bot \) and \( a \vee a^\star = \top \); + \item a \emph{Boolean algebra}, if it is distributive, bounded, and complemented. + \end{itemize} +\end{definition} +\begin{remark} + \begin{enumerate} + \item Distributive lattices model the fragment of a deduction system with only the conjunction and disjunction operators. + Boolean algebras model classical propositional logic. + \item Every lattice has an ordering, defined by \( a \leq b \) when \( a \wedge b = a \). + This ordering models the provability relation between propositions. + \end{enumerate} +\end{remark} +\begin{example} + \begin{enumerate} + \item Let \( I \) be a set. + The power set \( \mathcal P(I) \) can be made into a Boolean algebra by taking \( \wedge = \cap \) and \( \vee = \cup \). + \item More generally, let \( X \) be a topological space. + The set of closed and open sets of \( X \) form a Boolean algebra; they can also be thought of as the propositions in classical logic. + In fact, all Boolean algebras are of this form. + This result is known as Stone's representation theorem. + \item For any \( \mathcal L \)-structure \( \mathcal M \) and subset \( B \subseteq \mathcal M \), the set \( \qty{\varphi(\mathcal M) \mid \varphi(\vb x) \in \mathcal L_B} \) of definable subsets with parameters in \( B \) is a Boolean algebra. + \end{enumerate} +\end{example} + +\subsection{Filters} +\begin{definition} + Let \( X \) be a lattice. + A \emph{filter} \( \mathcal F \) on \( X \) is a subset of \( X \) such that + \begin{enumerate} + \item \( \mathcal F \neq \varnothing \); + \item \( \mathcal F \) is \emph{upward closed}: if \( f \leq x \) and \( f \in \mathcal F \) then \( x \in \mathcal F \); + \item \( \mathcal F \) is \emph{downward directed}: if \( x, y \in \mathcal F \), then \( x \wedge y \in \mathcal F \). + \end{enumerate} +\end{definition} +For property (ii), we might also say that \( \mathcal F \) is a \emph{terminal segment} of \( X \). +% large things get "stuck in the filter"; filters measure largeness