%%%%%%%%%%%%%%%%%%%%%%% file template.tex %%%%%%%%%%%%%%%%%%%%%%%%% % % This is a template file for the LaTeX package SVJour2 for the % Springer journal "Archive for Mathematical Logic". % % Springer Heidelberg 2004/12/07 % % Copy it to a new file with a new name and use it as the basis % for your article. Delete % as needed. % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % \documentclass[runningheads,numbook,draft]{svjour2} % \smartqed % flush right qed marks, e.g. at end of proof % \usepackage{graphicx} %\usepackage{biograph} % to allow for author biography at the end % \usepackage{mathptmx} % use Times fonts if available on your TeX system % % insert here the call for the packages your document requires %\usepackage{latexsym} % etc. \usepackage[polutonikogreek,english]{babel} \usepackage[oxonia]{psgreek} \usepackage{amsmath,amssymb} \usepackage{amscd} \usepackage{bm} \usepackage{pstricks} % for two figures \usepackage{upgreek} % for \upomega \usepackage{verbatim} % provides a comment environment %\usepackage{showlabels} % % please place your own definitions here and don't use \def but % \newcommand{}{} %%%%% General: \newcommand{\Gk}[1]{\selectlanguage{polutonikogreek}#1\selectlanguage{english}} \newcommand{\defn}[2]{\textbf{#1#2}} % for words being defined (the % second argument is for a plural % ending, punctuation, etc.) \renewcommand{\leq}{\leqslant} \renewcommand{\geq}{\geqslant} \renewcommand{\setminus}{\smallsetminus} \renewcommand{\phi}{\varphi} \newcommand{\eqcl}[1]{[\,#1\,]} % equivalence-class \newcommand{\tuple}[1]{\bm{#1}} % tuples \newcommand{\alg}[1]{#1^{\mathrm{alg}}} % algebraic closure \newcommand{\included}{\subseteq} % [the name suggests the meaning here] \newcommand{\pincluded}{\subset} % proper inclusion \newcommand{\gpgen}[1]{\langle#1\rangle} % group (or space) generated by #1 \newcommand{\size}[1]{\lvert#1\rvert} % cardinality \newcommand{\radix}[2][1]{\sqrt{\vphantom{#1}}#2} % square root % without vinculum \DeclareMathOperator{\trdeg}{tr-deg} % transcendence-degree \newcommand{\inv}{^{-1}} % multiplicative inverse \newcommand{\End}[1]{\operatorname{End}(#1)} % Endomorphism group %%%%% Quantifiers: \newcommand{\qsep}{\;} % follows a quantified variable \newcommand{\Forall}[1]{\forall{#1}\qsep } \newcommand{\Exists}[1]{\exists{#1}\qsep } \newcommand{\Existsunique}[1]{\exists!\,{#1}\qsep } %%%%% Other logical: \newcommand{\proves}{\vdash} \newcommand{\amp}{\;{}\mathbin{\&}{}\;} \newcommand{\Iff}{\iff} %%%%% General model-theory: \newcommand{\lang}{\mathcal{L}} % a language or signature \DeclareMathOperator{\modelclass}{Mod} \newcommand{\Mod}[1]{\modelclass^{\included}(#1)} % the category of % models with embeddings \newcommand{\Model}[1]{\modelclass^{\elsub}(#1)} % the category of % models with elementary embeddings \DeclareMathOperator{\diag}{diag} % (Robinson) diagram \newcommand{\str}[1]{\mathfrak{#1}} % structure \newcommand{\elsub}{\preccurlyeq} % is an elementary substructure of %%%%% Interpretations \newcommand{\intp}[2]{{#1}_{#2}} % interpretation of form. #1 w.r.t. #2 \newcommand{\iuni}[1]{\delta_{#1}{}} % universe for interpretation #1 \newcommand{\ifunc}[2]{f_{#1}^{#2}} % surj. map for interpretation % #1 in structure #2 %%%%% Specific theories: \newcommand{\VSp}[1]{\operatorname{VS}_{#1}} % vector-spaces with an % n-ary predicate for linear dependence \newcommand{\VSpm}[1]{\operatorname{VS}_{#1}^{\mathrm m}} % vector-spaces %of minumum dimension n with an n-ary predicate for linear dependence \newcommand{\VSpr}[1]{\operatorname{VS}_{#1}^{\mathrm r}} % reducts of % models of this \newcommand{\VSpst}[1]{{\VSp{#1}}\!^*} % model-companion for these %%%%% Specific structures: \newcommand{\stnd}[1]{\mathbb{#1}} \newcommand{\C}{\stnd{C}} % complex numbers \newcommand{\R}{\stnd{R}} % real numbers \newcommand{\Q}{\stnd{Q}} % rational numbers \newcommand{\Z}{\stnd{Z}} % integers \newcommand{\F}[1]{\stnd{F}_{#1}} % prime field of char. #1 \newcommand{\vnn}{\upomega} % von-Neumann natural numbers %%%%% Vectors: \newcommand{\vsp}{vector-space} % I like to hyphenate; others might not \newcommand{\vsps}{\vsp s} \newcommand{\Vsp}{Vector-space} \newcommand{\Vsps}{\Vsp s} \newcommand{\vect}[1]{\bm{#1}} % vectors \newcommand{\vx}{\vect x} \newcommand{\vy}{\vect y} \newcommand{\vz}{\vect z} \newcommand{\vw}{\vect w} \newcommand{\va}{\vect a} \newcommand{\vb}{\vect b} \newcommand{\svmult}{\ast} % multiplication of scalar and vector \newcommand{\crossprod}{\bm{\times}} % cross-product \newcommand{\ldep}[1]{P^{#1}} % predicate for linear dependence \newcommand{\transp}[1]{{#1}^{\mathrm t}} % transpose \newcommand{\Id}[1][n]{\mathrm{I}_{#1}} % identity matrix % \journalname{Archive for Mathematical Logic} % \begin{document} \title{Model-theory of \vsps\\ over unspecified fields } %\subtitle{Do you have a subtitle?\\ If so, write it here} \titlerunning{\Vsps} % if too long for running head \author{David Pierce } %\authorrunning{Short form of author list} % if too long for running head \institute{David Pierce \at Mathematics Department, Middle East Technical University, Ankara, Turkey, 06531 \\ \email{dpierce@metu.edu.tr} } \date{Received: date / Revised: date} % The correct dates will be entered by the editor \maketitle \begin{abstract} \Vsps{} over unspecified fields can be axiomatized as one-sorted structures, namely, abelian groups with the relation of parallelism. Parallelism is binary linear dependence. When equipped with the $n$-ary relation of linear dependence for some positive integer $n$, a \vsp{} is existentially closed if and only if it is $n$-dimensional over an algebraically closed field. In the signature with an $n$-ary predicate for linear dependence for \emph{each} positive integer $n$, the theory of infinite-dimensional \vsps{} over algebraically closed fields is the model-completion of the theory of \vsps. %Insert your abstract here. Include up to five keywords. \keywords{\vsp{} \and model-completeness \and sort} \subclass{03C10 \and 03C60} \end{abstract} \setcounter{section}{-1} \section{Introduction}\label{sect:intro} In Abraham Robinson's original definition \cite[\S2.2]{MR0472504}, a (first-order) theory $T$ is \defn{model-complete}{} if the theory $T\cup\diag{\str M}$ is complete whenever $\str M\models T$. Two equivalent formulations are as follows, in current notation, where $\str M$ and $\str N$ are models of $T$: \begin{enumerate} \item $\str M\included\str N\implies\str M\elsub_1\str N$ (\defn{Robinson's Test}{}) \cite[2.3.1]{MR0472504}; \item $\str M\included\str N\implies\str M\elsub\str N$ \cite[2.4.1]{MR0472504}. \end{enumerate} Model-completeness is of use in the discovery of complete theories, since a model-com\-plete theory with a model that embeds in every other model is complete \cite[4.1.6]{MR0472504}. Discovery of \emph{model}-complete theories---complete or not---is also of interest in itself. Robinson gives some examples of model-complete theories, including the theories of non-trivial torsion-free divisible abelian groups \cite[\S3.1]{MR0472504}, algebraically closed fields \cite[\S3.2]{MR0472504}, and real-closed fields \cite[\S3.3]{MR0472504}; these examples are now standard \cite[ch.~3]{MR1924282}. Robinson notes also, in effect, that the theory of non-trivial \vsps{} \emph{over a given field} is model-complete \cite[\S3.6]{MR0472504}. However, although his notation would seem to allow it, he does not appear to consider the theory of \vsps{} \emph{tout court} (over an unspecified field). %; this would have been a theory of \emph{two}-sorted structures. In the most usual formulation, a \vsp\ is a two-sorted structure. About early versions of model-theory, Angus Macintyre notes \cite[\S2.4, p.~198]{MR1988966}: \begin{quote} For no good reason, natural structures were forced into a one-sorted formulation (and by now it is a considerable nuisance that there is no basic model-theory text in a many-sorted setting). \end{quote} Nonetheless, the possibility of treating \vsps{} as two-sorted structures has been acknowledged in one textbook \cite[\S5.4]{MR1800596}; also, modules in general have been treated as two-sorted structures \cite[ch.~9]{MR1057608}. A natural signature for this treatment of vector-spaces includes: \begin{enumerate} \item the signature of abelian groups, for the vectors; \item the signature of rings, for the scalars; \item a symbol for multiplication of vectors by scalars (in the present paper, this symbol is $\svmult$). \end{enumerate} In this signature, let $T$ be the theory of \vsps, and if $n\in\{1,2,3,\dots\}\cup\{\infty\}$, let $T_n$ be the theory of \vsps{} of dimension $n$. Andrey Kuzichev \cite{MR1255099} gives an explicit elimination of quantified vector-variables in $T_n$. Indeed, suppose $C$ is a formula whose vector-variables are from the tuple $(x_1,\dots,x_m,y)$. (Kuzichev says $m\geq1$, but this is not necessary if summations $\sum_{i=1}^0f_i$ are understood to be $0$.) Let $C'$ be the result of replacing each $y$ in $C$ with $\sum_{i=1}^m\alpha_i\svmult x_i$, and let $C^*$ be the result of replacing each atomic sub-formula $\lambda\svmult y=\sum_{i=1}^m\mu_i\svmult x_i$ of $C$ with $\lambda=0\land\sum_{i=1}^m\mu_i\svmult x_i=0$. Then \begin{equation*} T\proves\Exists yC\leftrightarrow (\Exists{\tuple{\alpha}}C'\lor (C^*\land\Exists y\Forall{\tuple{\alpha}}y\neq\sum_{i=1}^m\alpha_i\svmult x_i)). \end{equation*} In each of the theories $T_n$, we can express the formula $\Exists y\Forall{\tuple{\alpha}}y\neq\sum_{i=1}^m\alpha_i\svmult x_i$ without quantified vector-variables. Kuzichev gives the corollary that, if $U$ is a complete theory of the scalar-field, then $T_n\cup U$ is complete. However, these complete theories $T_n\cup U$ are never model-complete, unless $n=1$. They are not even inductive: the union of a chain of models need not be a model (no matter what $U$ is, unless $n=1$; see \S\ref{sect:n} below). The problem is the lack of a way to express linear independence with an existential formula. The picture changes if predicates for linear independence or dependence are added. %\begin{wrapfigure}{r}{0.45\textwidth} \begin{figure} \centering \begin{pspicture} (0,0)(4,1.7) \psset{unit=5mm} \psline(0,3)(6,3)(6,1)(0,1)(0,3) \psline(6,1)(6,0)(3,0)(3,3) \psline(4,0)(4,3) \psline(3,0)(6,3) \uput[u](1.5,3){$x$} \uput[u](3.5,3){$y$} \uput[r](6,2){$x-y$} \end{pspicture} \caption{Euclid's II.5}\label{fig:euclid} \end{figure} %\end{wrapfigure} I arrive, myself, at the example of \vsps{} by noting first that, in the \emph{Elements} \cite{MR17:814b}, Euclid gives a geometric formulation of certain field-theoretic identities. For example, the identity $(x+y)(x-y)+y^2=x^2$ is Euclid's Proposition~II.5, but Euclid expresses it in terms of the squares and rectangles bounded by certain straight lines (\Gk{e>uje~iai gramma'i,} what we might call \emph{line-segments;} see Figure~\ref{fig:euclid}). At the beginning of the \emph{Geometry} \cite{Descartes-Geometry}, Ren\'e{} Descartes shows how straight lines (\emph{lignes droites}) can be multiplied to produce \emph{lines} rather than rectangles, if one line is chosen as unit: %\begin{wrapfigure}{l}{0.6\textwidth} \begin{figure} \centering \begin{pspicture} (0,0)(2.4,1.6) \psset{unit=4mm} \psline(0,0)(6,0)(1,4) \psline(1,0)(1.5,3.6) \psline(3,0)(3.3,2.16) \uput[ul](1,0){$D$} \uput[ul](3,0){$A$} \uput[ur](6,0){$B$} \uput[ur](1.5,3.6){$E$} \uput[ur](3.3,2.16){$C$} \end{pspicture} \caption{Descartes's geometric multiplication}\label{fig:descartes} \end{figure} %\end{wrapfigure} \begin{quote} \begin{itshape} Soit par exemple $AB$ l'vnit\'e, \& qu'il faille multiplier $BD$ par $BC$, ie n'ay qu'a ioindre les poins $A$ \& $C$, puis tirer $DE$ parallele a $CA$, \& $BE$ est le produit de cete Multiplication% \footnote{`For example, let $AB$ be taken as unity, and let it be required to multiply $BD$ by $BC$. I have only to join the points $A$ and $C$, and draw $DE$ parallel to $CA$; then $BE$ is the product of $BD$ and $BC$' \cite[p.~5]{Descartes-Geometry}.} [Fig.~\ref{fig:descartes}]. \end{itshape} \end{quote} By means of Descartes's observation, we can interpret the scalar-field in a \vsp{} of dimension at least two by means of the binary relation of \emph{parallelism,} that is, binary linear dependence (\S\ref{sect:parallel}). So \vsps{} of dimension at least two are axiomatizable, as one-sorted structures, in the signature of abelian groups with a binary predicate. Also, the axioms can be chosen to be $\forall\exists$. When the predicate for parallelism is used, then the existentially closed \vsps{} are two-dimensional. (It does not matter whether a separate sort for the scalars is retained or not.) If, more generally, an $n$-ary predicate for linear dependence is used, then the existentially closed \vsps{} are $n$-dimensional over algebraically closed fields (\S\ref{sect:n}). The class of such models being elementary, its theory is model-complete, although it does not admit full quantifier-elimination. On the value of looking at basic structures, I quote again Descartes, this time from near the end of the fourth of his \emph{Rules for the Direction of the Mind} \cite{Descartes-Rules}: \begin{quote} \begin{itshape} Quant \`a moi, conscient de ma faiblesse, j'ai d\'ecid\'e d'observer opini\^a\-trement, dans ma qu\^ete de connaissances, un ordre tel qu'en partant toujours des choses les plus simples et les plus faciles, je m'interdise de passer \`a d'autres, avant que dans les premi\`eres il ne m'apparaisse qu'il ne reste plus rien \`a d\'esirer; c'est pourquoi j'ai pouss\'e jusqu'\`a ce jour, aussi loin que j'ai pu, l'\'etude de cette math\'ematique universelle\dots% \footnote{`Aware how slender my powers are, I have resolved in my search for knowledge of things to adhere unswervingly to a definite order, always starting with the simplest and easiest things and never going beyond them till there seems to be nothing further which is worth achieving where they are concerned. Up to now, therefore, I have devoted all my energies to this universal mathematics\dots' \cite[p.~20]{Descartes-Eng}.} \end{itshape} \end{quote} The result about existentially closed \vsps\ means, in part, that a \vsp{} of $n+1$ dimensions can be embedded in an $n$-dimensional space so that linear independence of $n$-element sets (more precisely, $n$-tuples) is preserved. A similar result arises in the model-theory of fields: Let $(a_0,\dots,a_n)$ be algebraically independent over a field $K$, and let $(b,c)$ be a generic solution to $\sum_{i=0}^na_ix^i=y$. (This generalizes \cite[Example 8.1.2, p.~291]{MR1924282}; see also \cite[Example 2.1.8, p.~77]{MR1429864}.) Then $\trdeg(K(a_0,\dots,a_n)/K)=n+1$, while $\trdeg(K(a_0,\dots,a_n,b,c)/K(b,c))=n$, although every $n$-tuple of elements of $K(a_0,\dots,a_n)$ that is algebraically independent over $K$ is still independent over $K(b,c)$. Recently and independently, Moshe Kamensky \cite{Kamensky} has worked with (two-sorted) vector-spaces and has given theories of them that have elimination of quantifiers; this elimination is achieved by means of additional sorts, as for the Grassmannians of the finite powers of the scalar-field. \section{Interpretation of the field in the \vsp{}}\label{sect:parallel} In a chapter of his own textbook on the subject, Robin Hartshorne \cite[ch.~3]{MR1761093} presents Descartes's approach to geometry. It is perhaps a Cartesian spirit that allows Alfred Tarski \cite{MR0106185,MR1791303} to axiomatize Euclidean geometry by means of the ternary relation of between-ness and the quaternary relation of equidistance; these relations allow the underlying (real-closed) field to be recovered from the space. In the present section, I specify an origin in space and work with the group-structure determined by it; then only a binary relation is needed to recover the underlying (arbitrary) field. This observation was useful to me in showing that a field equipped with a space and Lie-ring of derivations can be described as a one-sorted structure in two ways: as a field with certain operators, or as a Lie-ring with certain operators. A \defn{\vsp}{} can be defined as a triple $(V,K,\svmult)$, where $V$ is an abelian group (of \defn{vector}{s}), $K$ is a field (of \defn{scalar}{s}), and~$\svmult$ is a function $(x,\vect v)\mapsto x\svmult \vect v$ from $K\times V$ to $V$ so that: \begin{enumerate} \item the functions $\vect v\mapsto a\svmult \vect v$ (where $a\in K$) are endomorphisms of $V$, and \item the function $x\mapsto (\vect v\mapsto x\svmult \vect v)$ is a ring-homomorphism from $K$ into $(\End V,\circ)$. \end{enumerate} Since $K$ is a field, the homomorphism into $(\End V,\circ)$ is an embedding, unless $V$ is trivial; excluding this case, we may treat the homomorphism as an inclusion, so that multiplication in $K$ is literally composition. Even in the trivial case, let us retain the symbol for composition to denote multiplication of scalars. Model-theoretically then, a \vsp{} is a two-sorted structure, in the signature $\{+,-,\vect0,\circ,0,1,\svmult\}$; here the elements of $\{+,-\}$ do double duty, taking arguments from the sort of vectors \emph{or} the sort of scalars. I shall use normal-face variables for scalars (as $0$), and boldface for vectors (as $\vect0$) or, in more general contexts, for tuples of variables. In every vector-space, the binary relation $\parallel$ on the sort of vectors is defined by the sentence \begin{equation}\label{eqn:pardef} \vx\parallel\vy\leftrightarrow\Exists u\Exists v(u\svmult\vx+v\svmult\vy=\vect0\land(u\neq0\lor v\neq0)). \end{equation} (Here and throughout the paper, outer universal quantifiers are suppressed.) The sentence \eqref{eqn:pardef} has the form \begin{equation}\label{eqn:form} \alpha\leftrightarrow\exists{z_0}\cdots\Exists{z_{n-1}}\beta, \end{equation} where $\alpha$ and $\beta$ are quantifier-free, and none of the $z_i$ appears in $\alpha$; such a sentence is equivalent to the conjunction of the $\forall\exists$-sentence $\exists{z_0}\cdots\Exists{z_{n-1}}(\alpha\to\beta)$ and the universal sentence $\beta\to\alpha$. Let $\VSp 2$ be the theory of \vsps{} that have been \emph{expanded} to the signature $\{+,-,\vect0,\circ,0,1,\svmult,\parallel\}$ so as to satisfy~\eqref{eqn:pardef}. (The subscript $2$ alludes to the number of arguments that $\parallel$ takes.) Models of the theory $\VSp 2$ can be \emph{extended} so as to satisfy also \begin{equation}\label{eqn:2} \Exists{\vx}\Exists{\vy}\vx\nparallel\vy. \end{equation} Let $\VSpm 2$ be the theory of the resulting structures: the \vsps{}, with parallelism, of dimension at least two. (The superscript `m' refers to a \emph{m}inimum dimension.) This theory is inductive. \defn{Reduction}{} of a one-sorted structure is the discarding of some named operations or relations; we may understand reduction of a many-sorted structure to involve also the discarding of some sorts. In this sense, every model $(V,K,\svmult,\parallel)$ of $\VSpm 2$ has a one-sorted reduct $(V,\parallel)$ in the signature $\{+,-,\vect0,\parallel\}$. This reduct is an abelian group with a certain binary relation; we might just call the reduct an \defn{abelian group with parallelism}. Let $\VSpr 2$ be the theory of these structures. The theories $\VSpm 2$ and $\VSpr 2$ will be related in Theorem~\ref{thm:V}. In general, the models of a theory $T$ in a signature $\lang$ are the objects of a category whose morphisms are either embeddings or elementary embeddings. Let the two possibilities for the category be $\Mod T$ and $\Model T$ respectively. If $\lang\included\lang'$, and $T'$ is a theory of $\lang'$, and reducts to $\lang$ of models of $T'$ are models of $T$, then we may speak of reduction from $T'$ to $T$: it is a functor $R$ from $\Model{T'}$ to $\Model{T}$, and from $\Mod{T'}$ to $\Mod{T}$ (this is a trivial case of Lemma~\ref{lem:int} below). Possibly, along with the reduction-functor $R$ from $\Model{T'}$ to $\Model{T}$, there is an \defn{expansion}-functor $E$ from $\Model{T}$ to $\Model{T'}$ such that $R\circ E$ is the identity, and for each model $\str A$ of $T'$, there is a `reasonable' isomorphism $\sigma_{ER}^{\str A}$ from $\str A$ to $E(R(\str A))$---reasonable in the sense that, for all models $\str A$ and $\str A^*$ of $T'$, for every elementary embedding $h$ of $\str A$ in $\str A^*$, the following diagram commutes: \begin{equation*} \begin{CD} \str A @>{h}>> \str A^*\\ @V{\sigma^{\str A}_{EF}}VV @VV{\sigma^{\str A^*}_{ER}}V\\ E(R(\str A)) @>>{E(R(h))}> E(R(\str A^*)) \end{CD} \end{equation*} In a word then, $R$ is an \defn{equivalence}{} of categories; we might say also that $R$ is a \defn{weakly conservative}{} reduction. For example, $R$ is weakly conservative in case $T'$ is $\VSp 2$, and $\lang$ is $\{+,-,\vect0,\circ,0,1,\svmult\}$ (so that $T$ is just the theory of vector-spaces). However, in this case, $E$ is not functorial on $\Mod{T}$. For example, $(\C,\R,\svmult)\included(\C,\C,\svmult)$, but $(\C,\R,\svmult,\parallel)\not\included(\C,\C,\svmult,\parallel)$. This is a point to be developed in \S\ref{sect:n}. Briefly, $E$ is not functorial here, because non-parallelism does not have an existential definition. In cases where $E$ \emph{is} functorial on $\Mod{T}$, and the diagram above commutes whenever $h$ embeds $\str A$ in $\str A^*$, so that $R$ is an equivalence of $\Mod{T'}$ and $\Mod{T}$,---in such cases, we may say that $R$ is simply a \defn{conservative}{} reduction. Now the statement of Theorem~\ref{thm:V} makes sense. For the proof, I first note that reduction and expansion are instances of \emph{interpretation.} In the account of Wilfrid Hodges \cite[\S5.3, p.~212]{MR94e:03002}, an \defn{interpretation}{} of a (one-sorted) structure $\str A$ whose signature is $\lang$ in a structure $\str B$ whose signature is $\lang'$ consists, for some positive integer $n$, of: \begin{enumerate} \item\label{item:form} a function $\phi\mapsto\intp{\phi}I$ converting each $k$-ary unnested atomic formula of $\lang$, for each $k$ in $\vnn$, into an $nk$-ary formula of $\lang'$, \item\label{item:univ} an $n$-ary formula $\iuni I$ of $\lang'$, and \item\label{item:func} a surjective function $\ifunc I{\str B}$ from $\iuni I^{\str B}$ to $A$, such that \begin{equation}\label{eqn:interpr} \str B\models\intp{\phi}I(\tuple b) \Iff \str A\models\phi(\ifunc I{\str B}(\tuple b)) \end{equation} for all $\tuple b$ from $\iuni I^{\str B}$. \end{enumerate} We may assume that the universe $A$ of $\str A$ is literally the set $\iuni I^{\str B}$, \emph{modulo} the equivalence-relation defined in $\str B$ by $\intp{(x=y)}I$. Then $\ifunc I{\str B}$ is the quotient-map, and $\str A$ is uniquely determined by $\str B$ and $\phi\mapsto\intp{\phi}I$ and $\iuni I$ \cite[5.3.3, p.~216]{MR94e:03002}: we may write $\str A=I(\str B)$. Thus we have a function $I$ converting structures of $\lang'$ into structures of $\lang$. In case $\str A$ has several sorts, composing a tuple $(A_{\alpha}\colon \alpha\in S)$ perhaps, then the definition must be modified: For each $\alpha$ in $S$, for some positive integer $n_{\alpha}$, there will be an formula $\iuni{I,\alpha}$ in some $n_{\alpha}$-tuple of variables (each variable being assigned to a sort), along with a function $\ifunc{I,\alpha}{\str B}$ from $\iuni{I,\alpha}^{\str B}$ onto $A_{\alpha}$, so that a variant of~\eqref{eqn:interpr} holds, namely, \begin{equation}\label{eqn:interpr2} \str B\models\intp{\phi}I(\tuple b_{\alpha(0)},\dots,\tuple b_{\alpha(k-1)}) \Iff \str A\models\phi(\ifunc{I,\alpha(0)}{\str B}(\tuple b_{\alpha(0)}),\dots, \ifunc{I,\alpha(k-1)}{\str B}(\tuple b_{\alpha(k-1)})) \end{equation} where $\tuple b_{\alpha(j)}\in\iuni{I,\alpha(j)}^{\str B}$. The function $\phi\mapsto\intp{\phi}I$ extends so that its domain comprises all unnested formulas of $\lang$, and \eqref{eqn:interpr} or~\eqref{eqn:interpr2} still holds \cite[5.3.2, p.~214]{MR94e:03002}: the extension takes $\phi\land\psi$ to $\intp{\phi}I\land\intp{\psi}I$, and $\lnot\phi$ to $\lnot(\intp{\phi}I)$, and $\Exists x\phi$ to $\Exists{(x^0,\dots,x^{n-1})}(\intp{\phi}I\land\iuni I(x^0,\dots,x^{n-1}))$ (with appropriate modifications in the many-sorted case). If there are theories $T$ of $\lang$ and $T'$ of $\lang'$ such that $I(\str B)\models T$ whenever $\str B\models T'$, then let us say that $I$ is an interpretation from $T'$ to $T$. In this case, we can understand $I$ as a functor from $\Model{T'}$ to $\Model T$, where, if $\str B$ and $\str B^{*}$ are models of $T'$, and $h$ is an elementary embedding of $\str B$ in $\str B^{*}$, then $I(h)$ is the well-defined elementary embedding of $I(\str B)$ in $I(\str B^{*})$ given by \begin{equation}\label{eqn:functor} I(h)(\ifunc I{\str B}(\tuple b))=\ifunc I{\str B^{*}}(h(\tuple b)) \end{equation} \cite[5.3.4(a), p.~217]{MR94e:03002}. If the formulas involved in $I$ are simple enough, then more follows: \begin{lemma}\label{lem:int} If $I$ is an interpretation from $T'$ to $T$ (and therefore a functor from $\Model{T'}$ to $\Model T$), then $I$ is also a functor from $\Mod {T'}$ to $\Mod T$, \emph{provided} both that the formula $\iuni I$ is existential, and also that the formula $\intp{\phi}I$ is existential whenever $\phi$ is an unnested atomic formula of $\lang$, or is $\lnot(x=y)$, or is $\lnot Rx^0\dotsb x^{k-1}$ for some predicate $R$ in $\lang$. \end{lemma} \begin{proof} The claim is a variant of \cite[5.3.4(b), p.~217]{MR94e:03002} and a refinement of \cite[\S5.4, exercise~3, p.~225]{MR94e:03002}. \end{proof} An interpretation $I$ from $T'$ to $T$, paired with an interpretation~$J$ from $T$ to $T'$, is a \defn{bi-interpretation}{} of $T'$ and $T$ if there are formulas $\chi_{IJ}$ of $\lang$, and $\chi_{JI}$ of $\lang'$, such that, whenever $\str A\models T$, then the set \begin{equation}\label{eqn:isomorphism} \{(a,\ifunc{I}{J(\str A)}(\ifunc{J}{\str A}(\tuple b_0),\dots, \ifunc{J}{\str A}(\tuple b_{n-1})))\colon \str A\models\chi_{IJ}(a,\tuple b_0,\dots,\tuple b_{n-1})\} \end{equation} is (the graph of) an isomorphism $\sigma^{\str A}_{IJ}$ from $\str A$ to $I(J(\str A))$, and similarly for all models of $T'$, with the places of $I$ and $J$ reversed (Hodges \cite[\S5.4(c), p.~222]{MR94e:03002}, generalizing Ahlbrandt and Ziegler \cite[p.~67]{MR831437}; in the case of several sorts, there will be several formulas $\chi_{IJ,\alpha}$, indexed with the sorts, and the graph of $\sigma_{IJ}^{\str A}$ will have components of the form $\{(a,\ifunc{I,\alpha}{J(\str A)} (\ifunc{J,\beta(0)}{\str A}(\tuple b_0),\dots, \ifunc{J,\beta(n_{\alpha}-1)}{\str A}(\tuple b_{n_{\alpha}-1})))\colon \str A\models\chi_{IJ,\alpha}(a,\tuple b_0,\dots,\tuple b_{n_{\alpha}-1})\}$). A special case was alluded to above, where reduction can be `undone' by an expansion. \begin{lemma}\label{lem:bi} If $(I,J)$ is a bi-interpretation of $T'$ and $T$, then $I$ is an equivalence of the categories $\Model{T'}$ and $\Model T$; if $I$ and $J$ are also functorial between $\Mod{T'}$ and $\Mod T$, then these categories too are equivalent, provided that the formulas $\chi_{IJ}$ and $\chi_{JI}$ are existential. \end{lemma} \begin{proof} To prove the first claim, if $\str A$ and $\str A^*$ are models of $T$, and $h$ is an elementary embedding of $\str A$ in $\str A^*$, then we show that the following diagram commutes (and likewise for models of $T'$): \begin{equation*} \begin{CD} \str A @>{h}>> \str A^*\\ @V{\sigma^{\str A}_{IJ}}VV @VV{\sigma^{\str A^*}_{IJ}}V\\ I(J(\str A)) @>>{I(J(h))}> I(J(\str A^*)) \end{CD} \end{equation*} Suppose $a\in A$ (and all models are one-sorted; extra sorts would be a challenge only to the notation.) Then (abbreviating the notation in~\eqref{eqn:isomorphism}) we have $\sigma_{IJ}^{\str A}(a)=\ifunc I{J(\str A)}(\ifunc J{\str A}(\tuple b))$ for some $\tuple b$ such that $\str A\models\chi_{IJ}(a,\tuple b)$. Then $\str A^*\models\chi_{IJ}(h(a),h(\tuple b))$ since $h$ is elementary, so by~\eqref{eqn:functor} \begin{multline*} \sigma_{IJ}^{\str A^*}(h(a)) =\ifunc I{J(\str A^*)}(\ifunc J{\str A^*}(h(\tuple b))) =\ifunc I{J(\str A^*)}(J(h)(\ifunc J{\str A}(\tuple b)))\\ =I(J(h))(\ifunc I{J(\str A)}(\ifunc J{\str A}(\tuple b)) =I(J(h))(\sigma_{IJ}^{\str A}(a)). \end{multline*} So the diagram does commute, and the first claim is proved. If $h$ is merely an embedding, but $I(h)$ is still well-defined as an embedding by~\eqref{eqn:functor}, then the same argument works, provided also $\chi_{IJ}$ is existential. \qed \end{proof} Now let us return to the special case of reductions and expansions. The following lemma singles out some of the basic properties of \vsps{} to be used in proving Theorem~\ref{thm:V}. \begin{lemma}\label{lem:basic} Suppose $(V,K,\svmult,\parallel)\models\VSpm 2$. \begin{enumerate} \item\label{item:p1} Say $\vx,\vy\in V$, and $\vx\neq0$. Then \begin{equation*} \vx\parallel\vy\Iff\Exists uu\svmult\vx=\vy\Iff \Existsunique uu\svmult\vx=\vy. \end{equation*} \item\label{item:p2} The relation $\parallel$, restricted to $V\setminus\{\vect0\}$, is an equivalence-relation. \item\label{item:p3} The set $\{\vy\in V\colon \vx\parallel\vy\}$ is a subgroup of $V$, for all $\vx$ in $V$. \item\label{item:p4} Let $\vx$, $\vz$, and $\vw$ be vectors in $V$ such that $\vx\nparallel\vz$; let $u\in K$; and let $\vy=u\svmult\vx$. Then \begin{equation*}%\label{eqn:-} u\svmult\vz=\vw\Iff \vz\parallel\vw\amp \vx-\vz\parallel\vect y-\vw. \end{equation*} \end{enumerate} \end{lemma} \begin{proof} I write a proof for item~\ref{item:p4} only, assuming items~\ref{item:p1},~\ref{item:p2}, and~\ref{item:p3}. %See Fig.~\ref{fig:parallel}. Assume $\vx\nparallel\vz$ and $\vy=u\svmult\vx$. Part of item~\ref{item:p3} is that $\vect0$ is parallel to every vector; so $\vz\neq\vect0$. Similarly, but with the help of item~\ref{item:p2}, we have $\vz\nparallel\vx-\vz$, so $\vx-\vz\neq\vect0$. Hence, if $u\svmult\vz=\vw$, then $\vz\parallel\vw$ by item~\ref{item:p1}, and also $\vy-\vw=u\svmult(\vx-\vz)$, which is parallel to $\vx-\vz$ for the same reason. Conversely, say $\vz\parallel\vw$. Then $\vw=t\svmult\vz$ for some $t$ in $K$, so that $\vy-\vw=u\svmult\vx-t\svmult\vz=u\svmult(\vx-\vz)+(u-t)\svmult\vz$. If also $\vx-\vz\parallel\vy-\vw$, then $(u-t)\svmult\vz\parallel\vx-\vz$ by item~\ref{item:p3}, so $u=t$. \qed\end{proof} \begin{theorem}\label{thm:V} Reduction from $\VSpm 2$ to $\VSpr 2$ is conservative. \end{theorem} \begin{proof} We have $R$ from $\Mod{\VSpm 2}$ to $\Mod{\VSpr 2}$. To go the other way, we want to interpret an arbitrary model $(V,K,\svmult,\parallel)$ of $\VSpm 2$ in its reduct, $(V,\parallel)$, obtaining an expansion-functor $E$. (It will follow that the reducts $(V,\parallel)$ compose an elementary class.) In the notation for interpretations given earlier, we shall need formulas $\iuni{E,\mathrm{vec}}$ and $\iuni{E,\mathrm{sca}}$; these will be $\vx=\vx$ and \begin{equation*} \vx_0\neq\vect 0\land\vx_0\parallel\vx_1, \end{equation*} respectively. Then $\ifunc{E,\mathrm{vec}}{(V,\parallel)}$ will be the identity on $V$, while $\ifunc{E,\mathrm{sca}}{(V,\parallel)}$ will be given by \begin{equation}\label{eqn:function} \ifunc{E,\mathrm{sca}}{(V,\parallel)}(\vect a_0,\vect a_1)=b \Iff (V,K,\svmult,\parallel)\models b\svmult\vect a_0=\vect a_1. \end{equation} Indeed, the function $\ifunc{E,\mathrm{sca}}{(V,\parallel)}$ is well defined on $\iuni{E,\mathrm{sca}}^{(V,\parallel)}$, and is surjective onto $K$, by Lemma~\ref{lem:basic} (item \ref{item:p1}): \begin{enumerate} \item If $(V,\parallel)\models\iuni{E,\mathrm{sca}}(\vect a_0,\vect a_1)$, then there is a unique $b$ in $K$ such that $(V,K,\svmult,\parallel)\models b\svmult\vect a_0=\vect a_1$. \item If $b\in K$, and $\vect a\neq\vect0$, then $(V,\parallel)\models\iuni{E,\mathrm{sca}}(\vect a,b\svmult\vect a)$. \end{enumerate} Let $\ifunc{E,\mathrm{sca}}{(V,\parallel)}$ be denoted by \begin{equation*}%\label{eqn:f} (\vx,\vy)\mapsto\eqcl{\vx:\vy}. \end{equation*} We have to define the function $\phi\mapsto\intp{\phi}E$. This will be the identity at all formulas with no scalar-variables. It is now enough to suppose that $\phi$ is one of \begin{equation*} x=y,\quad x+y=z,\quad x\circ y=z,\quad x\svmult\vy=\vz. \end{equation*} The formula $\intp{(x=y)}E$ will be the equation \begin{equation}\label{eqn:classes} \eqcl{\vx_0:\vx_1}=\eqcl{\vy_0:\vy_1}, \end{equation} expressed in $\{+,-,0,\parallel\}$. To obtain this expression, first note that \begin{equation}\label{eqn:mult-int} \vy_0\neq\vect0\amp \eqcl{\vx_0:\vx_1}\svmult\vy_0=\vy_1\Iff \eqcl{\vx_0:\vx_1}=\eqcl{\vy_0:\vy_1}. \end{equation} %The converse of \eqref{eqn:mult-int} also holds by definition. Now let $\psi(\vx_0,\vx_1,\vy_0,\vy_1)$ denote the formula \begin{equation*} \vx_0\nparallel\vy_0\land\vx_0\parallel\vx_1\land\vy_0\parallel\vy_1 \land\vx_0-\vy_0\parallel\vx_1-\vy_1. \end{equation*} (The situation can be illustrated by Fig.~\ref{fig:descartes} of \S\ref{sect:intro}, with $(A,B,C,D,E)$ replaced with $(\vx_0,\vect 0, \vy_0,\vx_1,\vy_1)$.) This formula $\psi$ defines a subset of $\iuni{E,\mathrm{sca}}^{(V,\parallel)}\times \iuni{E,\mathrm{sca}}^{(V,\parallel)}$; that is, it defines a binary relation on $\iuni{E,\mathrm{sca}}^{(V,\parallel)}$. This relation is symmetric, though not reflexive, and not quite transitive. However, by Lemma~\ref{lem:basic} (item~\ref{item:p4}) and \eqref{eqn:mult-int}, and by the assumption that $\dim_KV\geq2$, we have \begin{multline*} \eqcl{\vx_0:\vx_1}=\eqcl{\vy_0:\vy_1}\amp \vx_0\nparallel\vy_0\Iff \psi(\vx_0,\vx_1,\vy_0,\vy_1);\\ \shoveleft{\eqcl{\vx_0:\vx_1}=\eqcl{\vy_0:\vy_1}\amp \vx_0\parallel\vy_0\Iff \vx_0\parallel\vy_0\amp\Exists{\vz_0}\Exists{\vz_1}}\\ (\psi(\vx_0,\vx_1,\vz_0,\vz_1)\amp \psi(\vz_0,\vz_1,\vy_0,\vy_1)). \end{multline*} From these observations, we obtain~\eqref{eqn:classes} as a formula of $\{+,-,0,\parallel\}$. A slight simplification is possible: $\intp{(x=y)}E$ can be the existential formula \begin{equation}\label{eqn:ker} \Exists{\vz_0}\Exists{\vz_1} (\psi(\vx_0,\vx_1,\vy_0,\vy_1)\lor (\psi(\vx_0,\vx_1,\vz_0,\vz_1)\land\psi(\vz_0,\vz_1,\vy_0,\vy_1))). \end{equation} We obtain the remaining formulas $\intp{\phi}E$ by means of \eqref{eqn:mult-int} and the following identities: \begin{gather*} \eqcl{\vx:\vy}+\eqcl{\vx:\vz}=\eqcl{\vx:\vy+\vz};\\ \eqcl{\vy:\vz}\circ\eqcl{\vx:\vy}=\eqcl{\vx:\vz}. \end{gather*} In particular, writing \eqref{eqn:classes} as an abbreviation for \eqref{eqn:ker}, we can tabulate the results: \begin{equation*} \begin{array}{c|c} \phi & \intp{\phi}E\\\hline x+y=z & \Exists{\vw}(\eqcl{\vx_0:\vw}=\eqcl{\vy_0:\vy_1}\land \eqcl{\vx_0:\vx_1+\vw}=\eqcl{\vz_0:\vz_1})\\ x\circ y=z & \Exists{\vw}(\eqcl{\vy_1:\vw}=\eqcl{\vx_0:\vx_1}\land \eqcl{\vy_0:\vw}=\eqcl{\vz_0:\vz_1})\\ x\svmult\vy=\vz & \eqcl{\vx_0:\vx_1}=\eqcl{\vy:\vz}\lor(\vy=\vect0\land\vz=\vect0). \end{array} \end{equation*} These formulas $\intp{\phi}E$ are existential. Also, $\lnot(\intp{(x=y)}E)$ is equivalent to the existential formula \begin{equation*} \Exists{\vw}(\eqcl{\vx_0:\vx_1}=\eqcl{\vy_0:\vw}\land\vw\neq\vy_1). \end{equation*} Now we have $E$ as a functor from $\Mod{\VSpr 2}$ to $\Mod{\VSpm 2}$, by Lemma~\ref{lem:int}. The composition $R\circ E$ is the identity, and $\sigma_{RE}^{(V,\parallel)}$ is the identity, and $\chi_{RE}$ is $\vx=\vy$. For the other side, $\sigma_{ER}^{(V,K,\svmult,\parallel)}$ is the identity on $V$, while on $K$ it takes $y$ to the equivalence-class $\{(\vx_0,\vx_1)\colon \vx_0\neq\vect0\amp y\svmult\vx_0=\vx_1\}$; correspondingly, $\chi_{ER,\mathrm{vec}}$ is $\vx=\vy$, while $\chi_{ER,\mathrm{sca}}$ is $\vx_0\neq\vect0\amp y\svmult\vx_0=\vx_1$. By Lemma~\ref{lem:bi}, the proof is complete. \qed \end{proof} \section{Existentially closed \vsps{}}\label{sect:n} It was said in \S\ref{sect:intro} that, if $U$ is a \emph{complete} theory of fields, then the theory, in $\{+,-,\vect0,\circ,0,1,\svmult\}$, of $n$-dimensional \vsps{} whose scalar-fields are models of $U$ is complete, but not model-complete or even inductive, if $n>1$. Indeed, suppose $V$ is a \vsp{} with basis $(\vect b_0,\dots,\vect b_{n-1})$ over a model $K$ of $U$, and $n>1$. If $K\pincluded K'$, then the $\vect b_i$ span a \vsp{} $W$ over $K'$, but we can have $\vect b_0=a\svmult \vect b_1$ for some $a$ in $K'\setminus K$, so that $\dim_{K'}W\leq n-1$. Still, $(V,K,\svmult)$ is a {substructure} of $(W,K',\svmult)$. We can have $K'\models U$, and we can extend $W$ to an $n$-dimensional space $V'$ over $K'$. So $(V,K,\svmult)$ is a substructure of $(V',K',\svmult)$, and both are models of $T_n\cup U$. Continuing, we form a chain \begin{equation}\label{eqn:chain} (V,K,\svmult)\pincluded(V',K',\svmult)\pincluded\dotsb\pincluded (V^{(k)},K^{(k)},\svmult)\pincluded\dotsb \end{equation} of models of $T_n\cup U$; the union of this chain has dimension {at most} $n-1$. The chain \emph{can} have dimension one, even if $n=\infty$: for each $k$, let $V^{(k)}$ have basis $(\vect b_k,\vect b_{k+1},\dots)$, and assume $\vect b_k=a_k\svmult\vect b_{k+1}$ for some $a_k$ in $K^{(k+1)}\setminus K^{(k)}$. However, when the structures in \eqref{eqn:chain} expand to models of $\VSpm 2$, they no longer form a chain: non-parallelism is not preserved in super-structures. Nonetheless, we shall see that there are chains where non-parallelism \emph{is} preserved in going up, but the union has dimension two. The conclusion will be that the existentially closed models of $\VSpm 2$ are two-dimensional. There is a generalization involving chains that preserve $n$-ary linear independence. If $n>0$, let $\ldep n$ be an $n$-ary predicate for linear dependence. (So $\ldep 2\vx\vy$ means $\vx\parallel\vy$; but $\ldep 1$ is superfluous, since $\ldep 1\vx$ is equivalent to $\vx=\vect0$.) For each positive $n$, there is a theory $\VSp n$ of \vsps{} in the signature $\{+,-,\vect0,\circ,0,1,\svmult,\ldep n\}$, axiomatized by the usual vector-space axioms, along with \begin{gather} \label{eqn:P} \ldep n\vx_0\dotsb\vx_{n-1}\leftrightarrow \exists{y^0}\cdots\Exists{y^{n-1}} \Big( \sum_{in$, so that there is a linearly independent $(n+1)$-tuple $(\vect a^0,\dots,\vect a^n)$ of vectors in $V$. Then $(V,K,\ldep n)$ has no solution to \begin{equation}\label{eqn:dep} \sum_{i=0}^nx^i\svmult\vect a_i=\vect0\land\bigvee_{i=0}^nx^i\neq0. \end{equation} But analyse $V$ as $V_0\oplus V_1$, where $V_0$ is spanned by the vectors $\vect a^i$. By Theorem~\ref{thm:cross}, there is a model $(W,L,\ldep n)$ of $\VSp n$ in which $(V_0,K,\ldep n)$ embeds, but which has a solution to \eqref{eqn:dep}. Then $(V,K,\ldep n)$ embeds in $(W\oplus L\otimes_KV_1,L,\ldep n)$, and the latter has a solution to \eqref{eqn:dep}. So again $(V,K,\ldep n)$ was not existentially closed. So the existentially closed models of $\VSp n$ are models of $\VSpst n$; in particular, every model of $\VSp n$ embeds in a model of $\VSpst n$. If $(V,K,\ldep n)$ is an $n$-dimensional model of $\VSp n$, then $\VSpst n\cup\diag{(V,K,\ldep n)}$ is complete, by a categoricity argument as in the proof of Theorem~\ref{thm:1}. Therefore $\VSpst n$ is the model-completion of the theory of $n$-dimensional models of $\VSp n$. However, $\VSpst n$ is not the model-completion of $\VSpm n$ (much less of $\VSp n$). Indeed, let $(a_0,\dots,a_n)$ be an $(n+1)$-tuple of algebraic numbers such that the $(n+2)$-tuple $(a_0,\dots,a_n,1)$ is linearly independent over $\Q$. By Theorem~\ref{thm:cross}, the row-spaces of the matrices \begin{equation*} \left( \begin{array}{c} \Id\\\hline \begin{matrix} a_0 & \cdots & a_{n-1} \end{matrix} \end{array}\right) \quad\text{ and }\quad \left( \begin{array}{c} \Id\\\hline \begin{matrix} a_1 & \cdots & a_n \end{matrix} \end{array}\right) \end{equation*} are substructures $\str A$ and $\str B$ of $((\alg{\Q})^n,\alg{\Q},\ldep n)$. Being $(n+1)$-dimensional, they are isomorphic models of $\VSpm n$; but no automorphism of $((\alg{\Q})^n,\alg{\Q},\ldep n)$ takes $\str A$ to $\str B$. \qed\end{proof} We can let $\VSp{\infty}$ be the union of the theories $\VSp n$; so it is the theory of \vsps{} in the signature $\{+,-,\vect0,\circ,0,1,\svmult,\ldep 1,\ldep 2,\ldep 3,\dots\}$. Let $\VSpst{\infty}$ be the theory, in the same signature, of infinite-dimensional \vsps{} over algebraically closed fields. \begin{theorem}\label{thm:infty} The theory $\VSpst{\infty}$ is the model-completion of $\VSp{\infty}$, but does not admit quantifier-elimination. \end{theorem} \begin{proof} Every model $(V,K,\ldep 1,\dots)$ of $\VSp{\infty}$ embeds in the model \begin{equation*} (\alg K\otimes_KV\oplus(\alg K)^{\vnn},\alg K,\ldep 1,\dots) \end{equation*} of $\VSpst{\infty}$. Also, let $(V,K,\ldep 1,\dots)\models\VSp{\infty}$, and $\kappa=\aleph_0+\size V+\size K$. Then $\VSpst{\infty}\cup\diag(V,K,\ldep 1,\dots)$ is complete, since all of its models $(W,L,\ldep 1)$ are isomorphic, provided $\trdeg(L/K)=\kappa$ and $\dim_L(W/(L\otimes_KV))=\kappa$; but these are just the models of size $\kappa$ that realize certain types. So $\VSpst{\infty}$ is the model-completion of $\VSp{\infty}$. Finally, the example at the end of the proof of Theorem~\ref{thm:1} shows that $\VSpst{\infty}$ does not admit elimination of quantifiers. Indeed, $(\gpgen{1,\radix 2},\Q)$ expands to a substructure of a model of $\VSpst{\infty}$ in which $1\parallel\radix 2$, but $\VSpst{\infty}\cup\diag(\gpgen{1,\radix 2},\ldep 1,\ldep 2,\dots)$ is not complete, since it does not specify which scalars ensure that $1\parallel\radix 2$. \qed\end{proof} \begin{theorem} The completions of the model-complete theories $\VSpst n$ are obtained by specifying a characteristic for the scalar field. The completions are $\vnn$-stable. \end{theorem} \begin{proof} Let $\VSpst{n,p}$ be the theory of models of $\VSpst n$ whose scalar fields have characteristic $p$ (positive or zero). If $n$ is finite, then $\VSpst{n,p}$ is uncountably categorical, hence complete and $\vnn$-stable. All models of $\VSpst{\infty,p}$ have a substructure isomorphic to $(\{\vect0\},\F p)$, where $\F p$ is a prime field of characteristic $p$. Hence $\VSpst{\infty,p}\proves\diag(\{\vect 0\},\F p)$. But here $(\{\vect0\},\F p)\models\VSp{\infty}$, so the theory $\VSpst{\infty}\cup\diag(\{\vect 0\},\F p)$ is complete by Theorem~\ref{thm:infty}. Therefore $\VSpst{\infty,p}$ is complete. Finally, let $(V,K)$ be a countable, \emph{definably closed} substructure of a big model of $\VSpst{\infty}$. This implies that, if some vectors in $V$ are linearly dependent, then scalars witnessing this can be found in $K$. A complete type of $\vx$ over $(V,K)$ says either that $\vx$ is linearly independent from $V$, or---for some $\vect a_i$ in $V$---that $\vx=\sum_{i