  In the paper we study inverse semigroups $\mathscr{B}(G)$,
$\mathscr{B}^+(G)$, $\bar{\mathscr{B}}(G)$ and $\bar{\mathscr{B}}\,^+(G)$ which
are generated by partial monotone injective translations of a positive cone of
a linearly ordered group $G$. We describe Green's relations on the semigroups
$\mathscr{B}(G)$, $\mathscr{B}^+(G)$, $\bar{\mathscr{B}}(G)$ and
$\bar{\mathscr{B}}\,^+(G)$, their bands and show that they are simple, and
moreover the semigroups $\mathscr{B}(G)$ and $\mathscr{B}^+(G)$ are bisimple.
We show that for a commutative linearly ordered group $G$ all non-trivial
congruences on the semigroup $\mathscr{B}(G)$ (and $\mathscr{B}^+(G)$) are
group congruences if and only if the group $G$ is archimedean. Also we describe
the structure of group congruences on the semigroups $\mathscr{B}(G)$,
$\mathscr{B}^+(G)$, $\bar{\mathscr{B}}(G)$ and $\bar{\mathscr{B}}\,^+(G)$.
