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We study the semigroup    of monotone injective partial selfmaps of the set of integers having cofinite domain and image. We show that    is bisimple and all of its non-trivial semigroup homomorphisms are either isomorphisms or group homomorphisms. We also prove that every Baire topology    on   , such that    is a Hausdorff semitopological semigroup, is discrete and we construct a non-discrete Hausdorff inverse semigroup topology    on   . We show that the discrete semigroup    cannot be embedded into some classes of compact-like topological semigroups and that its remainder under the closure in a topological semigroup  S  is an ideal in  S .

On monoids of monotone injective partial selfmaps of integers with cofinite domains and images