From λ-hollow Frames to λ-repletions in W: II. λ-repletions in W

Authors

Richard N. Ball, University of Denver
Anthony W. Hager, Wesleyan University
Joanne Walters-Wayland, Chapman UniversityFollow

Document Type

Article

Publication Date

1-27-2025

Abstract

In this article we analyze the fine structure of the essential extensions of an object of W, the category of divisible archimedean lattice ordered groups with designated weak units. In particular, we show that an object Ghas an ordinally indexed sequence {τ^α_G}_δG of essential extensions with the following features. τ⁰_G is (isomorphic to) the identity function on G.

• For every α>0, τ^α_G is an essential extension of G into a W-object which is of the form RLfor some frame L, and which is λ-replete for some λ.

• Every such extension is (isomorphic to) τ^α_G for a unique α.

• τ^δG_G is (isomorphic to) the maximal essential extension of G.

• If λ≤ν≤δG then τ^ν_G factors through τ^λ_G.

Here a W-object is said to be λ-replete if it has the following equivalent properties.

• Every λ-generated W-kernel is a polar.

• Every proper λ-generated W-kernel of Gis contained in a proper polar.

• For λ-generated W-kernels Ki, if K1 K2then there exists K3 such that 0=K3⊆K1and K3∩K2=0.

• For W-kernels K1⊆K2, if K1is λ-generated then K⊥⊥1⊆K2.

• Every W-kernel of Gis λ-closed, i.e., closed under λ-joins.

• Every W-homomorphism out of Gis λ-complete.

Comments

This article was originally published in Topology and Its Applications, volume 363, in 2025. https://doi.org/10.1016/j.topol.2025.109233

Recommended Citation

R.N. Ball, A.W. Hager, J. Walters-Wayland, From λ-hollow frames to λ-repletions in W: II. λ-repletions in W, Topol. Appl. 363 (2025). https://doi.org/10.1016/j.topol.2025.109233

Peer Reviewed

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