Semi-boolean and Yosida $\ell$-groups, Martinez and Yosida frames, and the $G+B$ construction

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 $\ell$-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.