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

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 G has an ordinally indexed sequence ${\left\{{\tau }_G^{\alpha }\right\}}_{{\delta }_G}$ of essential extensions with the following features.

• •
  ${\tau }_G^0$ is (isomorphic to) the identity function on G.
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  For every $\alpha >0$, ${\tau }_G^{\alpha }$ is an essential extension of G into a W-object which is of the form $RL$ for some frame L, and which is λ-replete for some λ.
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  Every such extension is (isomorphic to) ${\tau }_G^{\alpha }$ for a unique α.
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  ${\tau }_G^{{\delta }_G}$ is (isomorphic to) the maximal essential extension of G.
• •
  If $\lambda \le \nu \le {\delta }_G$ then ${\tau }_G^{\nu }$ factors through ${\tau }_G^{\lambda }$.

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

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  Every λ-generated W-kernel is a polar.
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  Every proper λ-generated W-kernel of G is contained in a proper polar.
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  For λ-generated W-kernels $K_i$, if $K_1⊈K_2$ then there exists $K_3$ such that $0\ne K_3\subseteq K_1$ and $K_3\cap K_2=0$.
• •
  For W-kernels $K_1\subseteq K_2$, if $K_1$ is λ-generated then $K_1^{\perp \perp }\subseteq K_2$.
• •
  Every W-kernel of G is λ-closed, i.e., closed under λ-joins.
• •
  Every W-homomorphism out of G is λ-complete.

We refer to this as the sequence of λ-repletions of G.