arXiv subjects

Anthony W. Hager

Publications and source records attributed to Anthony W. Hager.

4 recordsLinked to original sources

arXiv2026

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

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.

math.GR↗

arXiv2024

Archimedean l-groups with strong unit: cozero-sets and coincidence of types of ideals

$\bf{W}^*$ is the category of the archimedean l-groups with distinguished strong order unit and unit-preserving l-group homomorphisms. For $G \in \bf{W}^*$, we have the canonical compact space $YG$, and Yosida representation $G \leq C(YG)$, thus, for $g \in G$, the cozero-set coz(g) in $YG$. The ideals at issue in $G$ include the principal ideals and polars, $G(g)$ and $g^{\perp \perp}$, respectively, and the $\bf{W}^*$-kernels of $\bf{W}^*$-morphisms from $G$. The ``coincidences of types" include these properties of $G$: (M) Each $G(g) = g^{\perp \perp}$; (Y) Each $G(g)$ is a $\bf{W}^*$-kernel; (CR) Each $g^{\perp \perp}$ is a $\bf{W}^*$-kernel (iff each coz(g) is regular open). For each of these, we give numerous ``rephrasings", and examples, and note that (M) = (Y) $\cap$ (CR). This paper is a companion to a paper in preparation by the present authors, which includes the present thrust in contexts less restrictive and more algebraic. Here, the focus on $\bf{W}^*$ brings topology to bear, and sharpens the view.

math.GR↗

arXiv2024

Sufficiently many projections in archimedean vector lattices with weak order unit

The property of a vector lattice of sufficiently many projections (SMP) is informed by restricting attention to archimedean $A$ with a distinguished weak order unit $u$ (the class, or category, $\bf{W}$), where the Yosida representation $A \leq D(Y(A,u))$ is available. Here, $A$ SMP is equivalent to $Y(A,u)$ having a $π$-base of clopen sets of a certain type called ``local". If the unit is strong, all clopen sets are local and $A$ is SMP if and only if $Y(A,u)$ has clopen $π$-base, a property we call $π$-zero-dimensional ($π$ZD). The paper is in two parts: the first explicates the similarities of SMP and $π$ZD; the second consists of examples, including $π$ZD but not SMP, and constructions of many SMP's which seem scarce in the literature.

math.GN↗

arXiv2014

Pointfree pointwise suprema in unital archimedean $\ell$-groups

We generalize the concept of the pointwise supremum of real-valued functions to the pointfree setting. The concept itself admits a direct and intuitive formulation which makes no mention of points. But our aim here is to investigate pointwise suprema of subsets of $\mathcal{R}L$, the family of continuous real valued functions on a locale, or pointfree space. Our setting is the category $\mathbf{W}$ of archimedean lattice-ordered groups ($\ell$-groups) with designated weak order unit, with morphisms which preserve the group and lattice operations and take units to units. A main result is the appropriate analog of the Nakano-Stone Theorem: a (completely regular) locale $L$ has the feature that $\mathcal{R}L$ is conditionally pointwise complete ($σ$-complete), i.e., every bounded (countable) family from $\mathcal{R}L$ has a pointwise supremum in $\mathcal{R}L$, iff $L $ is boolean (a $P$-locale). We adopt a maximally broad definition of unconditional pointwise completeness ($σ$-completeness): a divisible $\mathbf{W}$-object $G$ is pointwise complete ($σ$-complete) if it contains a pointwise supremum for every subset which has a supremum in any extension. We show that the pointwise complete ($σ$-complete) $\mathbf{W}$-objects are those of the form $\mathcal{R}L$ for $L$ a boolean locale ($P$-locale). Finally, we show that a $\mathbf{W}$-object $G$ is pointwise $σ$-complete iff it is epicomplete.

math.GN↗