The class of semi-boolean <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\ell $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ℓ</mi> </math></EquationSource> </InlineEquation>-groups was introduced in 1968 by A. Bigard. These are the <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\ell $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ℓ</mi> </math></EquationSource> </InlineEquation>-groups G in which the principal convex <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\ell $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ℓ</mi> </math></EquationSource> </InlineEquation>-subgroup G(a) generated by any <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$a \in G$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>∈</mo> <mi>G</mi> </mrow> </math></EquationSource> </InlineEquation> is equal to the polar <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$a^{\perp \perp }$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>a</mi> <mrow> <mo>⊥</mo> <mo>⊥</mo> </mrow> </msup> </math></EquationSource> </InlineEquation>. Examples include all hyperarchimedean <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\ell $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ℓ</mi> </math></EquationSource> </InlineEquation>-groups and all existentially closed abelian <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\ell $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ℓ</mi> </math></EquationSource> </InlineEquation>-groups. Ordered by inclusion, the set of convex <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\ell $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ℓ</mi> </math></EquationSource> </InlineEquation>-subgroups of a semi-boolean <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\ell $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ℓ</mi> </math></EquationSource> </InlineEquation>-group is a Martínez frame (an algebraic frame with FIP in which every element is a d-element). Related are the Yosida <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$\ell $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ℓ</mi> </math></EquationSource> </InlineEquation>-groups, i.e., the <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$\ell $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ℓ</mi> </math></EquationSource> </InlineEquation>-groups whose frame of convex <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$\ell $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ℓ</mi> </math></EquationSource> </InlineEquation>-subgroups is a Yosida frame (an algebraic frame with FIP in which every compact element is a meet of maximal elements). Applying results on Martínez frames and Yosida frames, we obtain new characterizations of the semi-boolean and Yosida <InlineEquation ID="IEq17"> <EquationSource Format="TEX">$\ell $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ℓ</mi> </math></EquationSource> </InlineEquation>-groups, show that the former constitute a radical class and the latter do not, and present new examples with special properties. To build some of our examples, we introduce the <InlineEquation ID="IEq18"> <EquationSource Format="TEX">$G+B$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mo>+</mo> <mi>B</mi> </mrow> </math></EquationSource> </InlineEquation> construction for <InlineEquation ID="IEq19"> <EquationSource Format="TEX">$\ell $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ℓ</mi> </math></EquationSource> </InlineEquation>-groups, an adaptation of the <InlineEquation ID="IEq20"> <EquationSource Format="TEX">$A+B$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>+</mo> <mi>B</mi> </mrow> </math></EquationSource> </InlineEquation> construction from commutative algebra.

Semi-boolean and Yosida $\ell $-Groups, Martínez and Yosida frames, and the $G+B$ Construction

期刊论文
发表日期：

2026年2月4日

查看全文

Papiya Bhattacharjee,
Anthony W. Hager,
Warren Wm. McGovern,
Brian Wynne

摘要

The class of semi-boolean $\ell $ $ℓ$ -groups was introduced in 1968 by A. Bigard. These are the $\ell $ $ℓ$ -groups G in which the principal convex $\ell $ $ℓ$ -subgroup G(a) generated by any $a \in G$ $a \in G$ is equal to the polar $a^{\perp \perp }$ $a^{⊥ ⊥}$ . Examples include all hyperarchimedean $\ell $ $ℓ$ -groups and all existentially closed abelian $\ell $ $ℓ$ -groups. Ordered by inclusion, the set of convex $\ell $ $ℓ$ -subgroups of a semi-boolean $\ell $ $ℓ$ -group is a Martínez frame (an algebraic frame with FIP in which every element is a d-element). Related are the Yosida $\ell $ $ℓ$ -groups, i.e., the $\ell $ $ℓ$ -groups whose frame of convex $\ell $ $ℓ$ -subgroups is a Yosida frame (an algebraic frame with FIP in which every compact element is a meet of maximal elements). Applying results on Martínez frames and Yosida frames, we obtain new characterizations of the semi-boolean and Yosida $\ell $ $ℓ$ -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 $G+B$ $G + B$ construction for $\ell $ $ℓ$ -groups, an adaptation of the $A+B$ $A + B$ construction from commutative algebra.

Semi-boolean and Yosida \(\ell \)-Groups, Martínez and Yosida frames, and the \(G+B\) Construction

摘要