Semi-boolean and Yosida $\ell$-groups, Martinez and Yosida frames, and the $G+B$ construction
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912812666716160 |
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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 | The class of semi-boolean $\ell$-groups was introduced in 1968 by A. Bigard. These are the $\ell$-groups $G$ in which the principal convex $\ell$-subgroup $G(a)$ generated by any $a \in G$ is equal to the polar $a^{\perp \perp}$. Examples include all hyperarchimedean $\ell$-groups and all existentially closed abelian $\ell$-groups. Ordered by inclusion, the set of convex $\ell$-subgroups of a semi-boolean $\ell$-group is a \Mart frame (an algebraic frame with FIP in which every element is a $d$-element). Related are the Yosida $\ell$-groups, i.e., the $\ell$-groups whose frame of convex $\ell$-subgroups is a Yosida frame (an algebraic frame with FIP in which every compact element is a meet of maximal elements). Applying results on \Mart frames and Yosida frames, we obtain new characterizations of the semi-boolean and Yosida $\ell$-groups, show that the former constitute a radical class and the latter do not, and present new examples with special properties. To build some of our examples, we introduce the $G+B$ construction for $\ell$-groups, an adaptation of the $A+B$ construction from commutative algebra. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_11930 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Semi-boolean and Yosida $\ell$-groups, Martinez and Yosida frames, and the $G+B$ construction Bhattacharjee, Papiya Hager, Anthony W. McGovern, Warren Wm. Wynne, Brian Group Theory 06F20, 46E25, 06D22, 08C05, 54D80 The class of semi-boolean $\ell$-groups was introduced in 1968 by A. Bigard. These are the $\ell$-groups $G$ in which the principal convex $\ell$-subgroup $G(a)$ generated by any $a \in G$ is equal to the polar $a^{\perp \perp}$. Examples include all hyperarchimedean $\ell$-groups and all existentially closed abelian $\ell$-groups. Ordered by inclusion, the set of convex $\ell$-subgroups of a semi-boolean $\ell$-group is a \Mart frame (an algebraic frame with FIP in which every element is a $d$-element). Related are the Yosida $\ell$-groups, i.e., the $\ell$-groups whose frame of convex $\ell$-subgroups is a Yosida frame (an algebraic frame with FIP in which every compact element is a meet of maximal elements). Applying results on \Mart frames and Yosida frames, we obtain new characterizations of the semi-boolean and Yosida $\ell$-groups, show that the former constitute a radical class and the latter do not, and present new examples with special properties. To build some of our examples, we introduce the $G+B$ construction for $\ell$-groups, an adaptation of the $A+B$ construction from commutative algebra. |
| title | Semi-boolean and Yosida $\ell$-groups, Martinez and Yosida frames, and the $G+B$ construction |
| topic | Group Theory 06F20, 46E25, 06D22, 08C05, 54D80 |
| url | https://arxiv.org/abs/2410.11930 |