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Questions about the branch of algebra that deals with groups.

1 vote
1 answer
231 views

Subgroup of lattice-ordered group

Let $H$ be a subgroup of a lattice-ordered group $G$. Suppose that $H$ with the induced order is a lattice (but a priori not a sublattice), so that $H$ is a lattice-ordered group too. For $a, b\in H$, …
Rajnish's user avatar
  • 173
0 votes
1 answer
367 views

Lattice-ordered group of rational rank 1

Does there exist a lattice-ordered, not totally ordered, group of rational rank $1$? Rational rank 1 means isomorphic to a nonzero subgroup of $\mathbb{Q}$. There exist totally ordered groups of …
Rajnish's user avatar
  • 173
0 votes
2 answers
211 views

Direct limit of lattice-ordered groups

In general, any abelian group can be expressed as a direct limit of its f.g. subgroups. For the case of $\ell$-group (lattice-ordered group) is that true or not? As an abelian group we do not have pro …
Rajnish's user avatar
  • 173
1 vote
1 answer
182 views

Spliting of short exact exact sequences of partially ordered groups

Consider a short exact sequence of partially ordered groups $$0 \longrightarrow H \stackrel{\alpha}{\longrightarrow} G \stackrel{\beta} {\longrightarrow} G/H \longrightarrow 0 ,$$ where $H$ is a conve …
Rajnish's user avatar
  • 173