子代数格中的最小固有扩展

IF 0.6 4区 数学 Q3 MATHEMATICS
Anthony W. Hager, Brian Wynne
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引用次数: 0

摘要

设\({\mathcal{A}})是一类具有\(I,A\在{\math cal{A}}}中)的代数。我们解释了包含I的A的所有子代数的形式为\(S_{\mathcal{A}}}(I,A)\)的格中的格论“严格满足不可约/覆盖”情形\(B<;C\。对于群的类\({\mathcal{G}}),我们使用Beaumont和Zuckerman的不变量来确定\(s_。最后,我们证明了后者在具有强序单位和保单位群同态的阿基米德群的范畴\(\mathbf{W}^{*}\)中产生了一些(而不是全部)最小真本质扩张。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Minimum proper extensions in some lattices of subalgebras

Let \({\mathcal {A}}\) be a class of algebras with \(I, A \in {\mathcal {A}}\). We interpret the lattice-theoretic “strictly meet irreducible/cover” situation \(B < C\) in lattices of the form \(S_{{\mathcal {A}}}(I,A)\) of all subalgebras of A containing I, where we call such \(B < C\) a minimum proper extension (mpe), and show that this means B is maximal in \(S_{{\mathcal {A}}}(I,A)\) for not containing some \(r \in A\) and C is generated by B and r. For the class \({\mathcal {G}}\) of groups, we determine the mpe’s in \(S_{{\mathcal {G}}}(\{0\},{\mathbb {Q}})\) using invariants of Beaumont and Zuckerman and show that these (plus utilization of a Hamel basis) determine the mpe’s in \(S_{{\mathcal {G}}}(\{0\},{\mathbb {R}})\). Finally, we show that the latter yield some (not all) of the minimum proper essential extensions in \(\mathbf {W}^{*}\), the category of Archimedean \(\ell \)-groups with strong order unit and unit-preserving \(\ell \)-group homomorphisms.

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来源期刊
Algebra Universalis
Algebra Universalis 数学-数学
CiteScore
1.00
自引率
16.70%
发文量
34
审稿时长
3 months
期刊介绍: Algebra Universalis publishes papers in universal algebra, lattice theory, and related fields. In a pragmatic way, one could define the areas of interest of the journal as the union of the areas of interest of the members of the Editorial Board. In addition to research papers, we are also interested in publishing high quality survey articles.
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