### 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 $K$...

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.

In the present paper, we will show that the set of minimal elements of a full affine semigroup $A\hookrightarrow {\mathbb{N}}_{0}^{k}$ contains a free basis of the group generated by $A$ in ${\mathbb{Z}}^{k}$. This will be applied to the study of the group ${\text{K}}_{0}\left(R\right)$ for a semilocal ring $R$.

Convergent and fundamental sequences are studied in a half linearly cyclically ordered group G with the abelian increasing part. The main result is the construction of the Cantor extension of G.

There is proved that a convex maximal line in a median group $G$, containing 0, is a direct factor of $G$.