On bicyclic extension of linearly ordered groups

We study inverse semigroups which are generated by partial translations of a positive cone of a linearly ordered group. In case of the additive group of integers with usual linear order one of these semigroups is isomorphic to the bicyclic semigroup. We describe their Green's relations, structure of bands and congruences. We show that for an archimedean linearly ordered group all non-trivial congruences on the constructed semigroups are group congruences and give an example that it is not true for a non-archimedean linearly ordered group.