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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | https://arxiv.org/abs/2404.18752 |
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| _version_ | 1866911858015862784 |
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| author | Bhattacharjee, Papiya Hager, Anthony W. McGovern, Warren Wm. Wynne, Brian |
| author_facet | Bhattacharjee, Papiya Hager, Anthony W. McGovern, Warren Wm. Wynne, Brian |
| contents | $\bf{W}^*$ is the category of the archimedean l-groups with distinguished strong order unit and unit-preserving l-group homomorphisms. For $G \in \bf{W}^*$, we have the canonical compact space $YG$, and Yosida representation $G \leq C(YG)$, thus, for $g \in G$, the cozero-set coz(g) in $YG$. The ideals at issue in $G$ include the principal ideals and polars, $G(g)$ and $g^{\perp \perp}$, respectively, and the $\bf{W}^*$-kernels of $\bf{W}^*$-morphisms from $G$. The ``coincidences of types" include these properties of $G$: (M) Each $G(g) = g^{\perp \perp}$; (Y) Each $G(g)$ is a $\bf{W}^*$-kernel; (CR) Each $g^{\perp \perp}$ is a $\bf{W}^*$-kernel (iff each coz(g) is regular open). For each of these, we give numerous ``rephrasings", and examples, and note that (M) = (Y) $\cap$ (CR). This paper is a companion to a paper in preparation by the present authors, which includes the present thrust in contexts less restrictive and more algebraic. Here, the focus on $\bf{W}^*$ brings topology to bear, and sharpens the view. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_18752 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Archimedean l-groups with strong unit: cozero-sets and coincidence of types of ideals Bhattacharjee, Papiya Hager, Anthony W. McGovern, Warren Wm. Wynne, Brian Group Theory 06D99, 08C05, 54C40 $\bf{W}^*$ is the category of the archimedean l-groups with distinguished strong order unit and unit-preserving l-group homomorphisms. For $G \in \bf{W}^*$, we have the canonical compact space $YG$, and Yosida representation $G \leq C(YG)$, thus, for $g \in G$, the cozero-set coz(g) in $YG$. The ideals at issue in $G$ include the principal ideals and polars, $G(g)$ and $g^{\perp \perp}$, respectively, and the $\bf{W}^*$-kernels of $\bf{W}^*$-morphisms from $G$. The ``coincidences of types" include these properties of $G$: (M) Each $G(g) = g^{\perp \perp}$; (Y) Each $G(g)$ is a $\bf{W}^*$-kernel; (CR) Each $g^{\perp \perp}$ is a $\bf{W}^*$-kernel (iff each coz(g) is regular open). For each of these, we give numerous ``rephrasings", and examples, and note that (M) = (Y) $\cap$ (CR). This paper is a companion to a paper in preparation by the present authors, which includes the present thrust in contexts less restrictive and more algebraic. Here, the focus on $\bf{W}^*$ brings topology to bear, and sharpens the view. |
| title | Archimedean l-groups with strong unit: cozero-sets and coincidence of types of ideals |
| topic | Group Theory 06D99, 08C05, 54C40 |
| url | https://arxiv.org/abs/2404.18752 |