From the author’s introduction. “We prove that, roughly speaking, the \(\alpha\)-completion \(G^{i\alpha}\) is the least completion of an \(\ell\)- group G in which G is large and which is order-closed in every extension which it \((G^{i\alpha})\) is large. Numerous related completion results are proved; in addition, we obtain fairly detailed structural descriptions of \(G^{i\alpha}\) in the cases when G is completely distributive, archimedean or strongly projectable. This paper, whose purpose is to explicate the structure of \(G^{i\alpha}\), is motivated by the opinion that \(\alpha\)-completeness is a natural and important lattice property with interesting consequences for \(\ell\)-groups.
An \(\ell\)-convergence structure on an \(\ell\)-group G is obtained by deciding which filters \({\mathcal F}\) on G are to converge to which points \(g\in G\), written \({\mathcal F}\Rightarrow g\), subject only to the following requirements. If \({\mathcal M}\supseteq {\mathcal F}\Rightarrow g\) then \({\mathcal M}\Rightarrow g\), if \({\mathcal F},{\mathcal M}\Rightarrow g\) then \({\mathcal F}\cap {\mathcal M}\Rightarrow g\), if \({\mathcal F}\Rightarrow f\) and \({\mathcal M}\Rightarrow m\) then \({\mathcal F}^{-1}\Rightarrow f^{-1}\), \({\mathcal F}{\mathcal M}\Rightarrow fm\), \({\mathcal F}\vee {\mathcal M}\Rightarrow f\vee m\), and \({\mathcal F}\wedge {\mathcal M}\Rightarrow f\wedge m\), and finally \(\dot g\Rightarrow g\) (where \(\dot g\) designate the filter with the base \(\{\) \(Y\subseteq G:\) \(g\in Y\})\). If these requirements are met we say that \(\Rightarrow\) is an \(\ell\)-convergence structure on G and refer to (G,\(\Rightarrow)\) as an \(\ell\)-convergence group, abbreviated \(\ell c\)- group. \(\Rightarrow\) is Hausdorff if no filter converges to more than one point, order closed if \({\mathcal F}\Rightarrow f\) implies \(ocl_ G({\mathcal F})\Rightarrow f\), (where \(ocl_ G({\mathcal F})\) designate the filter with the base \(\{ocl_ G(F):\) \(F\in {\mathcal F}\}\), and \(ocl_ G(X)=\cap \{Y:\) \(X\subseteq Y\subseteq G\) and Y is order closed\(\}\) is the order closure of X), and convex if \({\mathcal F}\to f\) implies \({\mathcal F}^{\sim}\Rightarrow f\) (where \({\mathcal F}^{\sim}\) designate the filter with the base \(\{F^{\sim}:\) \(F\in {\mathcal F}\}\), and \(X^{\sim}=\{g\in G:\) \(x_ 1\leq g\leq x_ 2\) for \(x_ 1,x_ 2\in X\}\) is the convexification of X). A subset \(X\subseteq G\) is closed with respect to \(\Rightarrow\) provided \(X\in {\mathcal F}\Rightarrow f\) implies \(f\in X\). If a filter \({\mathcal F}\) and element x of G satisfy the condition \(\bigvee (F\wedge G)=\bigwedge (F\vee g)=g\) for all \(F\in {\mathcal F}\) if and only if \(g=x\), we say that \({\mathcal F}\alpha\)-converges to x and write \({\mathcal F}\to x.\)
We summarize some of the main results of the present paper in the following Theorem: For every \(\ell\)-group G there is an \(\ell\)-group \(G^{i\alpha}\) which has properties (a)-(b6) below. \(G^{i\alpha}\) is unique up to \(\ell\)-isomorphism over G with respect to property (a) and minimality with respect to (b1). (Minimality with respect to property (b1) means that \(G^{i\alpha}\) enjoys property (b1), and that \(G\leq K<G^{i\alpha}\) implies K lacks property (b1).) If G is completely distributive and normal valued then \(G^{i\alpha}\) is unique either with respect to property (a) and minimality with respect to property (b5), or with respect to property (a) and minimality with respect to the conjunction of properties (b2), (b4), and (c). If G is archimedean or strongly projectable then \(G^{i\alpha}\) is unique either with respect to property (a) and minimality with respect to property (b5), or with respect to property (a) and minimality with respect to the conjunction of properties (b3) and (b4). If G is archimedean then \(G^{i\alpha}\) is unique with respect to property (a) and minimality with respect to property (b6).
(a) G is large in \(G^{i\alpha}\) (i.e. \(G\leq G^{i\alpha}\) and each nontrivial convex \(\ell\)-subgroup of \(G^{i\alpha}\) meets G nontrivially). (b1) \(G^{i\alpha}\) is \(\alpha\)-complete. That is, \(G^{i\alpha}\) is closed with respect to \(\alpha\)-convergence on any \(\ell\)-group in which \(G^{i\alpha}\) is large. (b2) \(G^{i\alpha}\) is sup \({\mathcal Y}\) complete [see R. Ball, Trans. Am. Math. Soc. 259, 357-392 (1980; Zbl 0441.06015)]. (b3) \(G^{i\alpha}\) is laterally complete. (b4) \(G^{i\alpha}\) is Dedekind-MacNeille complete. (b5) \(G^{i\alpha}\) is order closed in each \(\ell\)-group in which it is large. (b6) \(G^{i\alpha}\) has no archimedean extension in which it is order dense and proper. (c) Each positive element of \(G^{i\alpha}\) is a disjoint supremum of special elements. - \(G^{i\alpha}\) is closed with respect to any order closed Hausdorff \(\ell\)-convergence on any extension in which \(G^{i\alpha}\) is large.”