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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 work characterizes semi-boolean ℓ-groups via Martinez frames and analyzes Yosida frames to compare two central frame-theoretic notions, (M) and (Y). It proves that (M) is a radical class and provides new constructions and examples, including the G+B construction which yields retracts with closely related root systems of prime subgroups. The paper also clarifies the relationship between (M) and (Y), showing that (M) can be strictly larger than (Y) and that additional hypotheses (such as emc) affect their interplay. The G+B construction is then leveraged to produce archimedean ℓ-groups in (M) with Yosida spaces that are not zero-dimensional, highlighting nuanced structural differences between these two families. Overall, the results advance the frame-theoretic characterization of semi-boolean and Yosida ℓ-groups and provide a versatile toolset for generating and analyzing new examples.

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.

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

TL;DR

This work characterizes semi-boolean ℓ-groups via Martinez frames and analyzes Yosida frames to compare two central frame-theoretic notions, (M) and (Y). It proves that (M) is a radical class and provides new constructions and examples, including the G+B construction which yields retracts with closely related root systems of prime subgroups. The paper also clarifies the relationship between (M) and (Y), showing that (M) can be strictly larger than (Y) and that additional hypotheses (such as emc) affect their interplay. The G+B construction is then leveraged to produce archimedean ℓ-groups in (M) with Yosida spaces that are not zero-dimensional, highlighting nuanced structural differences between these two families. Overall, the results advance the frame-theoretic characterization of semi-boolean and Yosida ℓ-groups and provide a versatile toolset for generating and analyzing new examples.

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 and the latter do not, and present new examples with special properties. To build some of our examples, we introduce the construction for -groups, an adaptation of the construction from commutative algebra.

Paper Structure

This paper contains 12 sections, 23 theorems, 13 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lattices
  4. General $\ell$-groups
  5. The categories $\mathbf{W}$ and $\mathbf{W}^*$
  6. Martínez and Yosida frames
  7. Characterizations and preservation of $\mathop{\mathrm{(M)}}\nolimits$
  8. $\mathop{\mathrm{(Y)}}\nolimits$ and $\mathop{\mathrm{(M)}}\nolimits \cap \mathop{\mathrm{(Arch)}}\nolimits$
  9. The $G+B$ Construction
  10. $G+B$ and $\mathop{\mathrm{(M)}}\nolimits$
  11. Statements and Declarations
  12. Funding Declarations

Key Result

Theorem 2.1

Let $(G,u) \in {\bf W}$. There is a W-morphism $\varphi: (G,u) \rightarrow (D(YG), {\bf 1})$ such that for any $p\in YG$ and closed $V\subseteq YG$ not containing $p$, there is some $a\in G$ such that $\varphi(a)(p)=0$ and $\varphi(a)(q)=1$ for all $q\in V$. Moreover, $YG$ is the unique compact Hau

Theorems & Definitions (58)

  • Theorem 2.1: The Yosida Embedding Theorem
  • Lemma 2.2
  • Theorem 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 48 more