Semi-boolean and Yosida $\ell$-groups, Martinez and Yosida frames, and the $G+B$ construction
Papiya Bhattacharjee, Anthony W. Hager, Warren Wm. McGovern, Brian Wynne

TL;DR
This paper explores semi-boolean and Yosida $ ext{l}$-groups through algebraic frames, providing new characterizations, examples, and introducing a novel $G+B$ construction for $ ext{l}$-groups.
Contribution
It offers new characterizations of semi-boolean and Yosida $ ext{l}$-groups, shows their radical properties, and introduces the $G+B$ construction for $ ext{l}$-groups.
Findings
Semi-boolean $ ext{l}$-groups form a radical class.
Yosida $ ext{l}$-groups do not form a radical class.
New examples with special properties are constructed.
Abstract
The class of semi-boolean -groups was introduced in 1968 by A. Bigard. These are the -groups in which the principal convex -subgroup generated by any is equal to the polar . Examples include all hyperarchimedean -groups and all existentially closed abelian -groups. Ordered by inclusion, the set of convex -subgroups of a semi-boolean -group is a \Mart frame (an algebraic frame with FIP in which every element is a -element). Related are the Yosida -groups, i.e., the -groups whose frame of convex -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 -groups, show that the former constitute a radical class…
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Taxonomy
TopicsAdvanced Algebra and Logic · Rings, Modules, and Algebras · Fuzzy and Soft Set Theory
