### A characterization of ordered groups by means of segments

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Let $K$ be a field with a Krull valuation $\left|\phantom{\rule{0.277778em}{0ex}}\right|$ and value group $G\ne \left\{1\right\}$, and let ${B}_{K}$ be the valuation ring. Theories about spaces of countable type and Hilbert-like spaces in [1] and spaces of continuous linear operators in [2] require that all absolutely convex subsets of the base field $K$ should be countably generated as ${B}_{K}$-modules.By [1] Prop. 1.4.1, the field $K$ is metrizable if and only if the value group $G$ has a cofinal sequence. We prove that for any fixed cardinality ${\aleph}_{\kappa}$, there exists a metrizable field $K$...

Pseudo $\u2606$-autonomous lattices are non-commutative generalizations of $\u2606$-autonomous lattices. It is proved that the class of pseudo $\u2606$-autonomous lattices is a variety of algebras which is term equivalent to the class of dualizing residuated lattices. It is shown that the kernels of congruences of pseudo $\u2606$-autonomous lattices can be described as their normal ideals.

In this expository article we use topological ideas, notably compactness, to establish certain basic properties of orderable groups. Many of the properties we shall discuss are well-known, but I believe some of the proofs are new. These will be used, in turn, to prove some orderability results, including the left-orderability of the group of PL homeomorphisms of a surface with boundary, which are fixed on at least one boundary component.

Let $\Delta $ and $H$ be a nonzero abelian linearly ordered group or a nonzero abelian lattice ordered group, respectively. In this paper we prove that the wreath product of $\Delta $ and $H$ fails to be affine complete.