1\)).}\label{eq:3} \end{equation} \end{corollary} \begin{corollary}\label{corlem1thGOEntrelGRL} Let \((G,{\leq_G})\)~be an ordered group. T.f.a.e. \begin{asparaenum} \item \(A\vda[{\rh[\mathrm s ]}]B\). \item\label{corlem1thGOEntrelGRL2} There is an integer~\(n\geq1\) such that for some elements~\(a^{(n)}\in A^{(n)}\) and~\(b^{(n)}\in B^{(n)}\), \(a^{(n)}\leq_Gb^{(n)}\) holds. \end{asparaenum} \end{corollary} \begin{proof} Suppose that there are \(a_1,\dots,a_k,b_1,\dots,b_\ell\in G\) such that $a^{(n)}=n_1a_1+\dots+n_ka_k\leq_Gb^{(n)}=m_1b_1+\dots+m_\ell b_\ell$ with integers~\(n_i,m_j\geq0\) such that \(n_1+\dots+n_k=m_1+\dots+m_\ell=n\). By the Riesz refining lemma \citep[see e.g.][Theorem XI-2.11]{CACM}, there are integers~$p_{ij}\geq 0$ such that $n_i=\sum_{j=1}^\ell p_{ij}$ for each~$i$ and $m_j=\sum_{i=1}^kp_{ij}$ for each~$j$, so that \cref{corlem1thGOEntrelGRL2} may be written \[ \ndsp\sum_{i=1}^k\sum_{j=1}^\ell p_{ij}(a_i-b_j)\leq_G0\text{ for some integers $p_{ij}\geq0$ not all zero.} \] The equivalence follows by \cref{lemmath2GOEntrelGRL0}. \end{proof} \subsection{The \grl freely generated by an ordered group}\label{sec:lorenz-cliff-dieud} As an application, we provide the following description for the \grl freely generated by an ordered group. \begin{theorem}\label{ithmgogrlfree} For every ordered group~\(G\) we can construct an \grl~\(H\) with a morphism~\(\mor\colon G\to H\) such that \(0\leq_H\mor(a)\) holds if and only if \(0\leq_Gna\) for some~\(n\geq1\). More precisely, \(H\) is the \grl freely generated by~\(G\) (in the sense of the left adjoint functor of the forgetful functor) and can be constructed as the Lorenzen group associated with the finest \si, characterised by: \({\mor(a_1)\vii\dots\vii\mor(a_k)}\leq_H\mor(b_1)\vuu\dots\vuu\mor(b_\ell)\) holds if and only if there are integers~\(n_1,\dots,n_k,m_1,\dots,m_\ell\geq 0\) with \(n_1+\cdots+n_k=m_1+\dots+m_\ell\geq1\) such that \(n_1a_1+\dots+n_ka_k\leq_Gm_1b_1+\dots+m_\ell b_\ell\). \end{theorem} \Cref{ithmgogrlfree} is in fact a reformulation of the following proposition, enriched with an account of \cref{th2GOEntrelGRL0,corlem1thGOEntrelGRL,sclosed}. \begin{proposition} Let \((G,{\leq_G})\)~be an ordered group. The Lorenzen group associated with the finest \si for~\(G\) is the \grl freely generated by~\((G,{\leq_G})\) (in the sense of the left adjoint functor of the forgetful functor). \end{proposition} \begin{proof} The finest monoid of ideals for \(G\) is the meet-monoid~$M$ freely generated by~\(G\), and its Grothendieck \grl~$H$ is the \grl freely generated by~$M$: therefore $H$~is the \grl freely generated by~\(G\) as a monoid, and therefore also as a group. \end{proof} \Cref{ithmgogrlfree} may be seen as a generalisation of the following corollary, the constructive core of the classical Lo\-ren\-zen-Clif\-ford-Dieu\-don\-né theorem. %c %: Corollary{corthmgogrlfree} \begin{corollary}[Lorenzen-Clifford-Dieudonné, see {\citealp[Satz~14 for the $\mathrm{s}$-\scentrel]{Lor1939}; \citealp[Theorem~1]{Cli1940}; \citealp[Section~1]{Die1941}}]\label{corthmgogrlfree} The ordered group~\((G,{\leq_G})\)~is embeddable into an \grl if and only if \begin{equation*} \text{\(0\leq_Gna\) implies \(0\leq_Ga\)\quad(\(a\in G\), \(n>1\)).} \tag{\ref{eq:3}}\label{re:eq:3} \end{equation*} \end{corollary} %--------- fin corollary ------------------------------- % \begin{proof} The condition is clearly necessary. \Cref{ithmgogrlfree} shows that it yields the injectivity of the morphism \hbox{$\mor\colon G\to H$} as well as the fact that $\mor(x)\leq_H \mor(y)$ implies $x\leq_Gy$. \end{proof} % \begin{comments} \phantomsection\label{comment-s-system} \begin{asparaenum} \item In each of the three references given in \cref{corthmgogrlfree}, the authors invoke a maximality argument for showing that $G$~embeds in fact into a direct product of linearly ordered groups. The goal of \citet[\S~4; {\citeyear{Lor1953}}]{Lor1950} is to avoid the necessarily nonconstructive reference to linear orders in conceiving embeddings into an \grl, and this endeavour culminates in \cref{corthGOEntrelGRL}. \item The reader will recognise Condition~\cref{re:eq:3} of $\rh[\!\mathrm s]$-closed\-ness of \cref{sclosed} in the condition of embeddability stated here. In fact, in his Ph.D.\ thesis \citeyearpar{Lor1939}, \citeauthor{Lor1939} proves \cref{corthmgogrlfree} as a side-product of his enterprise of generalising the concepts of multiplicative ideal theory to the framework of preordered groups. He is following the Prüfer approach presented in \cref{secGoembedsGrl}, in which $\rh[\!\mathrm s]$-closedness is being introduced according to \cref{def-closed} and the equivalence with Condition~\cref{re:eq:3} is easy to check (see \citealp[page~358]{Lor1939}, or \citealp[I, \S~4, Théorème~2]{Jaf1960}).\eoe \end{asparaenum} \end{comments} % \subsection{The regularisation of the system of Dedekind ideals}\label{subsecaritalaLor} Let us resume \cref{sec:forc-posit-an-1} with a crucial lemma. %l %: Lemma{lemLordivgroup} \begin{lemma}\label{lemLordivgroup} One has \(A\vda[{\rh[\mathrm d ]}]1\) if and only if \(\gen{A}_{\id[A]}\ni 1\). \end{lemma} %----------- fin lemma ----------------------------------- % \begin{proof} Suppose that $A\vda[{\rh[\mathrm d ]}]1 $, i.e.\ by \cref{forcing-dedekind} that there are elements $x_1,\dots,x_n\in G$ such that $\gen{A}_{\id[x_1^{\pm1},\dots,x_n^{\pm1}]}\ni 1$. It suffices to prove the following fact and to use it in an induction argument: suppose that $\gen{A}_{\id[A,x]}\ni 1$ and $\gen{A}_{\id[A,x^{-1}]}\ni 1$; then $\gen{A}_{\id[A]}\ni 1$. In fact, the hypothesis means that $\gen{A,Ax,\dots,Ax^p}_{\id[A]}\ni 1$ and $\langle A,Ax^{-1},\dots,Ax^{-p}\rangle_{\id[A]}\ni 1$ for some~$p\geq0$, which implies that \[ \forall k\in\lrb{-p..p}\quad \big\langle Ax^{-p},\dots,Ax^{-1},A,Ax,\dots,Ax^p\big\rangle_{\id[A]}\ni x^k\text, \] i.e.\ that there is a matrix~$M$ with coefficients in~$\gen{A}_{\id[A]}$ such that $M(x^k)_{-p}^p=(x^k)_{-p}^p$, i.e.\ $(1-M)(x^k)_{-p}^p=0$. Let us now apply the determinant trick: multiplying~$1-M$ by the matrix of its cofactors and expanding it yields that $\gen{A}_{\id[A]}\ni 1$. Conversely, let $a_1,\dots,a_k$ be the elements of~$A$. For each~$i$, $a_ia_i^{-1}=1$, so that $\gen{A}_{\id[a_i^{-1}]}\ni 1$ and $A\mathrel{{(\rhd_{\mathrm{d}})}_{a_1^{\pm1},\dots,a_k^{\pm1}}}1$ for every choice of signs with at least one negative sign: the only missing choice of signs consists in the hypothesis $\gen{A}_{\id[A]}\ni 1$. \end{proof} An element $b\in K$ is said to be \emph{integral over the ideal} $\gen{A}_\id$ when an integral dependence relation $b^p=\sum_{k=1}^{p} c_k b^{p-k}$ with $c_k\in{\gen{A}_\id}^{k}$ holds for some~$p\geq1$. If $A=\so1$, then this reduces to the same integral dependence relation with $c_k\in\id$, i.e.\ to $b$~being integral over~$\id$. Note that if $A$ contains nonintegral elements, i.e.\ elements not in~$\id$, then ${\gen{A}_\id}^2$ may or may not be contained in $\gen{A}_\id$: consider respectively e.g.\ the ideal $\gen{1,\frac ut}$ in $k[T,U]/(T^3-U^2)=k[t,u]$ and ideals in a Prüfer domain. \begin{theorem}[{\citealp[Satz~2]{Lor1953}}]\label{TLordivgroup} Let \(\id\) be an integral domain and \(\rh[\mathrm d]\) its system of Dedekind ideals. \begin{asparaenum} \item\label{TLordivgroup1} One has \(A\vda[{\rh[\mathrm d ]}]b\)---i.e.\ the element~\(b\) is \(\rh[\mathrm d]\)-dependent on~\(A\); there are \(x_1,\dots,x_n\) such that \( \gen{A}_{\id[x_1^{\pm1},\dots,x_n^{\pm1}]}\ni b\) for every choice of signs---if and only if \(b\)~is integral over the ideal~\(\gen{A}_\id\). \item\label{TLordivgroup2} One has\/ $A\vda[{\rh[\mathrm d ]}]B$---that is, there are\/ $x_1,\dots,x_n$ such that\/ $\gen{AB^{-1}}_{\id[x_1^{\pm1},\dots,x_n^{\pm1}]}\ni 1$ for every choice of signs---if and only if\/ $\sum_{k=1}^p{\langle AB^{-1}\rangle_\id}^{k}\ni 1$ for some\/ $p\geq1$, i.e.\ there is an equality\/ $\som_{k=1}^p f_k=1$ with each\/ $f_k$ a homogeneous polynomial of degree\/ $k$ in the elements of\/ $AB^{-1}$ with coefficients in\/ $\id$. \item\label{TLordivgroup3} The divisibility group~\(G\) is \(\rh[\!\mathrm d]\)-closed, i.e.\ the equivalence \[{a\vda[{\rh[\!\mathrm d]}]b} \enskip\iff\enskip a\text{ divides \(b\)}\] holds, if and only if \(\id\) is integrally closed. \end{asparaenum} \end{theorem} \begin{proof} \begin{asparaenum} \item[(\ref{TLordivgroup1}--\ref{TLordivgroup2})]\refstepcounter{enumi}\refstepcounter{enumi} This follows from the previous lemma because \[ \begin{aligned} A\vda[{\rh[\mathrm d ]}]b\enskip&\iff\enskip Ab^{-1}\vda[{\rh[\mathrm d ]}]1\text,\\ \ndsp\sum_{k=1}^{p} c_k b^{p-k}=b^p\text{ with }c_k\in{\gen{A}_\id}^{k}\enskip&\ndsp\iff\enskip\sum_{k=1}^p{\langle Ab^{-1}\rangle_\id}^{k}\ni 1\text,\\ \ndsp\gen A_{\id[A]}\ni 1\enskip&\ndsp\iff\enskip\exists p\geq1\enskip\sum_{k=1}^p{\gen{A}_\id}^{k}\ni 1\text. \end{aligned} \] \item[(\ref{TLordivgroup3})] $\rh[\!\mathrm d]$-closedness is equivalent to $1\vda[{\rh[\mathrm d ]}]b\implies \id\ni b$; by \cref{TLordivgroup1}, $1\vda[{\rh[\mathrm d ]}]b$ holds if and only if $b$~is integral over~$\id$.\qedhere \end{asparaenum} \end{proof} \subsection{The Lorenzen divisor group of an integral domain}\label{secalaLor} In this section, we note consequences of \cref{ithmDivLorsi,ithGOEntrelGRL} for Lorenzen's theory of divisibility presented in \cref{subsecaritalaLor}. %d %: Definition{defiLordivgroup} \begin{definition}\label{defiLordivgroup} Let $\id$~be an integral domain. The \emph{Lorenzen divisor group} $\mathrm{Lor}(\id)$ of $\id$ is the Lorenzen group associated by \cref{defLorgroup} with the system of Dedekind ideals~$\rh[\mathrm{d}]$ for the divisibility group of~$\id$. \end{definition} % ----------- fin definition -------------------------------- The following version of \cref{ithmDivLorsi,embedLorgroup} takes into account the informations provided by \cref{TLordivgroup}; \cref{thmDivLor2-1} emphasises the fact that a regular \entrel is characterised by its restriction to~$\Pfs(G)\times G$ (\cref{ABA-B} of \cref{cor+x-x}). %t %: Theorem{thmDivLor2} \begin{theorem}\label{thmDivLor2} Let \(\id\) be an integral domain with field of fractions~\(K\) and divisibility group \(G=K\eti/\id\eti\). The \entrel~\(\vda[{\rh[\!\mathrm d]}]\) generates the Lorenzen divisor group \(\mathrm{Lor}(\id)\) together with a morphism of ordered groups \(\mor\colon G\to\mathrm{Lor}(\id)\) that satisfies the following properties. % \begin{asparaenum} % \item\label{thmDivLor2-1} The \gui{ideal Lorenzen gcd} of \(a_1,\dots,a_k\in K\etl\) is characterised by \[ \mor(a_1)\vii\dots\vii \mor(a_k) \leq \mor (b)\enskip \iff \begin{aligned}[t] &b\text{ is integral over}\\ &\text{the ideal \(\gen{a_1,\dots,a_k}_\id\).} \end{aligned} \] \item\label{thmDivLor2-2} The morphism \(\mor\) is an embedding if and only if \(\id\) is integrally closed. % \end{asparaenum} \end{theorem} \Cref{thmDivLor2-1} lends itself to an extensional formulation in terms of the integral closure $\Icl_K(a_1,\dots,a_k)$ of ideals \(\gen{a_1\dots,a_k}_\id\) in the field of fractions~$K$. If \(a_1,\dots,a_k\in \id\etl\), i.e.\ if one considers integral finitely generated ideals, it seems more appropriate to find a formulation in terms of the integral closure $\Icl(a_1,\dots,a_k)$ in the integral domain. This works because the elements $a_1,\dots,a_k,b\in K\etl$ in a relation $a_1,\dots,a_k\vda[{\rh[\!\mathrm d]}]b$ may be translated by an~$x$ into~$\id\etl$. This yields the following theorem, in which we use the conventional additive notation for divisor groups of an integral domain. It takes into account the construction of the Lorenzen group as the Grothendieck \grl of the meet-monoid associated with the regularisation of the system of Dedekind ideals in the proof of \cref{ithGOEntrelGRL}, i.e.\ as formal differences $\Vii\mor(A)-\Vii\mor(B)$; we take advantage of the fact that $\Vii\mor(A)-\Vii\mor(B)=\Vii\mor(xA)-\Vii\mor(xB)$ for every~$x$, so that it suffices to use integral ideals in this construction. %c %: Corollary{corthmDivLor2} \begin{theorem}\label{corthmDivLor2} Let \(\id\) be an integral domain. The Lorenzen divisor group \(\mathrm{Lor}(\id)\) can be realised extensionally in the following way. % \begin{asparaitem} % \item A \emph{basic divisor} is realised as the integral closure \(\Icl(a_1\dots,a_k)\) of an ordinary, i.e.\ integral finitely generated ideal \(\gen{a_1\dots,a_k}_\id\) with \(a_1,\dots,a_k\in \id\etl\). % \item The neutral element of the group, i.e.\ the divisor~\(0\), is realised as \(\Icl({1})\). % \item The meet of two basic divisors is realised as \[ \Icl(a_1,\dots,a_k)\vii\Icl(b_1,\dots,b_\ell)=\Icl(a_1,\dots,a_k,b_1,\dots,b_\ell). \] % \item The sum of two basic divisors is realised as \[ \Icl(a_1,\dots,a_k)+\Icl(b_1,\dots,b_\ell)=\Icl(a_1b_1,\dots\dots,a_kb_\ell). \] % \item The order relation between basic divisors is realised as \[ \Icl(a_1,\dots,a_k)\leq \Icl(b_1,\dots,b_\ell) \enskip \iff\enskip \Icl(a_1,\dots,a_k)\supseteq \Icl(b_1,\dots,b_\ell). \] In particular, \(\Icl(a)\leq\Icl(b)\) holds if and only if \(b\) is integral over \(\gen{a}_\id\). % \item Every divisor is realised as the formal difference of two basic divisors. % \end{asparaitem} \end{theorem} % --------- fin corollary ------------------------------- \begin{remarks}\phantomsection\label{remthmDivLor2} \begin{asparaenum} \item This theorem holds without condition of integral closedness, but beware of the following fact: if some $b\in K\etl\setminus\id\etl$ is integral over~$\id$, then $\Icl_K(1)\ni b$ and $0\leq\mor(b)$; however $\Icl(1)=\id$ and $\mor(b)$ is realised as a nonbasic divisor. An example for this is $\id=\QQ[t^2,t^3]$, $b=\frac{t^3}{t^2}$, $\mor(b) = \mor(t^3) - \mor(t^2)$, $\Icl(t^3)=\gen{t^3,t^4}_\id$, $\Icl(t^2)=\gen{t^2,t^3}_\id$. \item If every positive divisor is basic, then one can show the domain to be Prüfer. \item When $\id$ is a Prüfer domain, the Lorenzen divisor group $\mathrm{Lor}(\id)$ coincides with the usual divisor group, the group of finitely generated fractional ideals defined by Dedekind and Kronecker. In fact, all finitely generated ideals are integrally closed in a Prüfer domain, so that $\Icl(a_1,\dots,a_k)=\gen{a_1,\dots,a_k}_\id$. \item The integral domain $\id=\QQ[t,u]$ is a gcd domain of dimension $\geq 2$, so that its divisibility group $G$ is an \grl. The domain $\id$ is not Prüfer and the Lorenzen divisor group is much greater than $G$: e.g.\ the ideal gcd of $t^3$ and $u^3$ in $\mathrm{Lor}(\id)$ corresponds to the integrally closed ideal $\gen{t^3,t^2u,tu^2,u^3}$, whereas their gcd in $\id\etl$ is $1$, corresponding to the ideal~$\gen{1}$. In this case, we see that $G$ is a proper quotient of $\mathrm{Lor}(\id)$.\eoe \end{asparaenum} \end{remarks} The following corollary concentrates upon the cancellation property holding in \grls. Note that the integral closure of an integral finitely generated ideal in an integrally closed integral domain is equal to its integral closure in the field of fractions. % : corollary{corthmDivLor} \begin{corollary}[see {\citealp[pages~108--109]{Macaulay}}]\label{corthmDivLor} Let \(\id\)~be an integrally closed integral domain. When \(\fa\)~is a finitely generated integral ideal~\(\gen{a_1,\dots,a_k}_\id\) with \(a_1,\dots,a_k\in \id\etl\), we let \(\overline\fa=\Icl(a_1,\dots,a_k)\)~be the integral closure of~\(\fa\). Then, if \(\fa\),~\(\fb\), and~\(\fc\) are nonzero finitely generated integral ideals, we have the cancellation \property \[ \overline{\fa\,\fb}\supseteq\overline{\fa\,\fc}\enskip\implies\enskip \overline\fb\supseteq \overline\fc. \] \end{corollary} This corollary is a key result for ``containment in the wider sense'' as considered by Leopold \citet{kronecker83} (see \citet{penchevre}, pages 36--37). H. S.~\citet{Macaulay} gives a proof based on the multivariate resultant. We may also deduce it as a consequence of Prüfer's \cref{thSIPrufer} (see \cref{lemDivLor2item2} of \cref{lemDivLor2}, compare \citealp[\S~6]{Pru1932}, \citealp[Nr.~46]{Kru35}). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{\Sis and Prüfer's theorem}\label{secGoembedsGrl} In this section, we account for another way to obtain the Lorenzen group associated with an \si for an ordered group (\cref{defLorgroup}). This way has historical precedence, as it dates back to \citeauthor{Lor1939}'s Ph.D.\ thesis \citeyearpar{Lor1939}, that builds on earlier work by \citet{Pru1932}. In the case of the system of Dedekind ideals, this approach provides another way of understanding the Lorenzen divisor group of an integral domain. \subsection{Prüfer's properties \texorpdfstring{$\Gamma$}{Gamma} and \texorpdfstring{$\Delta$}{Delta}} Let us now express cancellativity of the \mmonoid as a property of the \si itself (a.k.a.\ ``endlich arithmetisch brauchbar'', ``e.a.b.'', see \cref{remhist3}), as in \citealt[\S~3]{Pru1932}. %l %: Lemma{lemSIJaf} \begin{lemma}[Prüfer's \Property~$\Gamma$ of cancellativity]\label{lemSIJaf} Let \(\rh\)~be an \si for an ordered group~\(G\). The associated \mmonoid~\(M\) is cancellative, i.e.\ \(\Vii(A+X)=_M\Vii(B+X)\) implies \(\Vii A=_M\Vii B\), if and only if the following property holds: \begin{equation} A+ X\leq_{\rh}b+ X \enskip\implies\enskip A\rh b\text.\label{propGamma} \end{equation} This holds if and only if $ A+ X\leq_{\rh} X\implies A\rh 0$. \end{lemma} %----------- fin lemma ----------------------------------- % \begin{proof} The second implication, a particular case of the first one, implies the first one by equivariance. Let us work with the first implication. Cancellativity means that if $ A+ X\leq_{\rh} B+ X$, then $ A\leq_{\rh} B$. \Property~\cref{propGamma} is necessary: take~\hbox{$B=\so b$}. Let us show that it is sufficient. Assume \hbox{$ A+ X\leq_{\rh} B+ X$} and let $b\in B$. As $B\rh b$, we have ${ B+ X}\leq_{\rh}{b+ X}$, whence ${ A+ X}\leq_{\rh}{b+ X}$. So $A\rh b$. Since this holds for each $b\in B$, we get $ A\leq_{\rh} B$.\qedhere \end{proof} % \begin{remark}\label{remGamma} The original version of Prüfer's Property~$\Gamma$ states, for a set-theoretical star-ope\-ra\-tion $A\mapsto A_r$ on nonempty finitely enumerated subsets of~$G$ as considered in \cref{remJafsi2} of \cref{remJafsi}, the cancellation property $(A+X)_r\supseteq(B+X)_r\implies A_r\supseteq B_r$.\eoe \end{remark} Prüfer's \cref{thSIPrufer} will reveal the significance of the following definition. We shall check in \cref{propGRLor} that it agrees with \cref{ideflorrel}. \begin{definition}[Prüfer's \Property~$\Delta$ of integral closedness]\label{def-closed} Let $\rh$ be an \si for an ordered group~$G$. The group~$G$ is \emph{$\rh$-closed} if ${ X\leq_{\rh}b+ X}\implies{0\leq_G b}$. \end{definition} \begin{remark} The original version of Prüfer's Property~$\Delta$ states the cancellation property ${X_r\supseteq b+X_r}\implies{0\leq_G b}$.\eoe \end{remark} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Forcing cancellativity: Prüfer's theorem} When the monoid~$M$ in \cref{ThSIJaf} is not cancellative, it is possible to adjust the \si in order to straighten the situation. A priori, it suffices to consider the Grothendieck \grl of~$M$ (\cref{thGoembedsGrl}). But we have to see that this corresponds to an \si for~$G$, and to provide a description for it. The following theorem is a reformulation of Prüfer's theorem \citep[\S~6]{Pru1932}. We follow the proofs in \citealt[pages~42--43]{Jaf1960}. In fact, the language of \scentrels simplifies the proofs. Jaffard's statement corresponds to \cref{thSIPrufer1,thSIPrufer4}, and \cref{thSIPrufer2,thSIPrufer3} have been added by us. %t %: Theorem{thSIPrufer} \begin{theorem}[Prüfer's theorem]\label{thSIPrufer} Let \(\rh\) be an \si for an ordered group~\(G\). We define the relation \(\Vda\) between~\(\Pfs(G)\) and~\(G\) by \[ A\Vda b \enskip\equidef\enskip\exists X\in \Pfs(G)\enskip A+ X \leq_{\rh}b+ X\text. \] % \begin{asparaenum} % \item\label{thSIPrufer1} The relation \(\Vda\) is an \si for \(G\), and the associated \mmonoid\allowbreak \(M_\mathrm{a}\) (\cref{ThSIJaf}) is cancellative. % \item\label{thSIPrufer2} The \mmonoid \(M_\mathrm{a}\) embeds into its Grothendieck \grl~\(H_\mathrm{a}\). % \item\label{thSIPrufer3} The system~\(\Vda\) is the finest \si~\(\rh'\) coarser than~\(\rh\) such that \(M_\mathrm{a}\) is cancellative, i.e.\ forcing \[ A+ X\leq_{\rh'}b+ X \enskip\implies\enskip A\rh' b\text. \] % \item\label{thSIPrufer4} The implication \(a\Vda b\implies a\leq_G b\) holds if (and only if) \(G\)~is \(\rh\)-closed (\cref{def-closed}); in this case, \(G\) embeds into~\(H_\mathrm{a}\). % \end{asparaenum} \end{theorem} %----------- fin theorem ----------------------------- \begin{proof} Note that if $ A+ X\leq_{\rh}b+ X$, then $ A+ X+ Y\leq_{\rh}b+ X+ Y$ for all~$Y$ (see the proof of \cref{ThSIJaf} on \cpageref{proofThSIJaf}). This makes the definition of~$\Vda$ very easy to use. In the proof below, we have two preorder relations on~$\Pfs(G)$ ($\leq_{\rh}$ \hbox{and $\leq_\mathrm{a}$}), and we shall proceed as if they were order relations (i.e.\ we shall descend to the quotients). \begin{asparaenum} \item[(\ref{thSIPrufer1})] \leavevmode\enspace\Bullet\enspace\emph{Reflexivity and \presord} (of the relation $\Vda$). Setting $X=\so{0}$ in the definition of~$\Vda$ shows that $a\leq_G b$ implies $a\Vda b$. \item[\Bullet\enspace\emph{Monotonicity}.] It suffices to note that the elements~${(A,A')+X}$ and~${A+X},{A'+X}$ of $\Pfs(G)$ are the same: therefore, if $ A+ X\leq_{\rh}b+ X$, then $ (A,A')+ X \leq_{\rh}b+ X$. \item[\Bullet\enspace\emph{Transitivity}.] Assume $A \Vda c$ and $A,c\Vda b$: we have an $X$ such that $ A+ X\leq_{\rh}c+ X$ and a $Y$ such that $ (A,c)+ Y\leq_{\rh}b+ Y$; these inequalities imply respectively $ A+X+Y\leq_{\rh} c+X+Y$ and $ A+X+Y, c+X+Y\leq_{\rh} b+X+Y$; we deduce $ A+X+Y\leq_{\rh} b+X+Y$, so that $A\Vda b$. \item[\Bullet\enspace\emph{\Equivariance}.] If $A\Vda b$, we have an $X$ such that $ A+ X\leq_{\rh}b+ X$, so that, since $\leq_{\rh}$~is \equivariant, $x+ A+ X\leq_{\rh}x+b+ X$. This yields $x+ A\Vda x+b$. \item[\Bullet\enspace\emph{Cancellativity} (of the \mmonoid\/~$M_\mathrm{a}$).] Let us denote by~$\leq_\mathrm{a}$ the order relation associated to~$\Vda$. By \cref{lemSIJaf}, it suffices to suppose that $ A+ X\leq _\mathrm{a} X$ and to deduce that $A\Vda 0$. But the hypothesis means that $A+X\Vda x$ for each $x\in X$, i.e.\ that for each $x\in X$ there is a $Y_x$ such that $ A+ X+ Y_x\leq_{\rh}x+ Y_x$. Let $Y=\sum_{x\in X}Y_x$: we have $ A+ X+ Y\leq_{\rh}x+ Y$. As $x\in X$~is arbitrary, $ A+ X+ Y\leq_{\rh} X+ Y$: this yields $A\Vda 0$ as desired. \item[(\ref{thSIPrufer2})] Follows from \cref{thSIPrufer1} by \cref{thGoembedsGrl}. \item[(\ref{thSIPrufer3})] This is immediate from the definition of~$\Vda$: it has been defined in a minimal way as coarser than~$\rh$ and forcing the cancellativity of the monoid $M_\mathrm{a}$ as characterised in \cref{lemSIJaf}. \item[(\ref{thSIPrufer4})] If $a\Vda b$, then we have an $X$ such that $a+ X\leq_{\rh}b+ X$, so that by a translation $ X\leq_{\rh}(b-a)+ X$. The hypothesis on~$G$ yields $0\leq_G b-a$. By a translation, we get $a\leq_G b$.\qedhere \end{asparaenum} \end{proof} % \begin{comment} This is the approach proposed in \citealt[\S~4]{Lor1939}. Lorenzen abandoned it in favour of \cref{ideflorrel} for the purpose of generalising his theory to noncommutative groups. See also \cref{remhist3,comment-s-system}.\eoe \end{comment} % d %: Definition{defiGRLor} \begin{definition}[see {\citealp[page~546]{Lor1939}}, or {\citealp[II, \S~2, 2]{Jaf1960}}]\label{defiGRLor} Let \(\rh\) be an \si for an ordered group~\(G\). The \grl in \cref{thSIPrufer2} of \cref{thSIPrufer} is the \emph{Lorenzen group} associated with~$\rh$. \end{definition} %----------- fin definition -------------------------------- %: Proposition{propGRLor} \begin{proposition}[{\citealp[Satz~27]{Lor1950}}]\label{propGRLor} The definition of \(A\Vda0\) in \cref{thSIPrufer} agrees with \cref{ideflorrel} of \(A\vda[\rh]0\). So \cref{def-closed} of \(\rh\)-closedness agrees with that of \cref{ideflorrel}, and \cref{defiGRLor} of the Lorenzen group agrees with that of \cref{defLorgroup}. \end{proposition} %----------- fin proposition ----------------------------- \begin{proof} This proposition expresses that, given an \si~$\rh$ for an ordered group $G$ and \hbox{an $A\in\Pfs(G)$}, we have $A\vda[\rh]0$ (\cref{ideflorrel}) if and only if ${A+X} \leq_{\rh}X$ for some~$X\in \Pfs(G)$. First, $A+Y \leq_{\rh_{\!x}} Y$ and $A +Z\leq_{\rh_{\!-x}} Z$ imply $A+X \leq_{\rh}X$ for some~$X$. In fact, we have~$p$ and~$q$ such that \[ \begin{aligned} A+Y, A+Y + x, \dots, A+Y + px &\leq_{\rh} Y\text{ and}\\ A+Z, A+Z - x, \dots, A+Z - qx&\leq_{\rh} Z\text{ hold,} \end{aligned} \] which yield that for $z\in Z$, $j\leq q$, $y\in Y$, and $k\leq p$, \[ \begin{aligned} A+Y+z-jx,\dots,A+Y+z+(p-j)x&\leq_{\rh} Y+z-jx\text{ and}\\ A+y+Z+kx,\dots,A+y+Z+(k-q)x&\leq_{\rh} y+Z+kx\text{ hold,} \end{aligned} \] so that $A + X \leq_{\rh} X$ for $X = Y+Z+\{ -qx,\dots, px \}$. In the other direction, assume that $A + X \rh x_i$ for each $x_i$ in $X=\so{x_1,\dots,x_m}$. Let $x_{i,j}={x_i-x_j}$ ($i