The projectable hull of an archimedean $\ell$-group with weak unit | ||
| Categories and General Algebraic Structures with Applications | ||
| مقاله 11، دوره 7، Special Issue on the Occasion of Banaschewski's 90th Birthday (II) - شماره پیاپی 1، مهر 2017، صفحه 165-179 اصل مقاله (1.14 M) | ||
| نوع مقاله: Research Paper | ||
| نویسندگان | ||
| Anthony W. Hager1؛ Warren Wm. McGovern2 | ||
| 1Department of Mathematics and CS, Wesleyan University, Middletown, CT 06459. | ||
| 2H. L. Wilkes Honors College, Florida Atlantic University, Jupiter, FL 33458. | ||
| چکیده | ||
| The much-studied projectable hull of an $\ell$-group $G\leq pG$ is an essential extension, so that, in the case that $G$ is archimedean with weak unit, ``$G\in {\bf W}$", we have for the Yosida representation spaces a ``covering map" $YG \leftarrow YpG$. We have earlier \cite{hkm2} shown that (1) this cover has a characteristic minimality property, and that (2) knowing $YpG$, one can write down $pG$. We now show directly that for $\mathscr{A}$, the boolean algebra in the power set of the minimal prime spectrum $Min(G)$, generated by the sets $U(g)=\{P\in Min(G):g\notin P\}$ ($g\in G$), the Stone space $\mathcal{A}\mathscr{A}$ is a cover of $YG$ with the minimal property of (1); this extends the result from \cite{bmmp} for the strong unit case. Then, applying (2) gives the pre-existing description of $pG$, which includes the strong unit description of \cite{bmmp}. The present methods are largely topological, involving details of covering maps and Stone duality. | ||
تازه های تحقیق | ||
Dedicated to Bernhard Banaschewski on the Occasion of his 90th Birthday, Communicated by Themba Dub | ||
| کلیدواژهها | ||
| Archimedean $l$-group؛ vector lattice؛ Yosida representation؛ minimal prime spectrum؛ principal polar؛ projectable؛ principal projection property | ||
| مراجع | ||
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