  Based on a notion by Gray and Kambites of hyperbolicity in the setting of
semimetric spaces like digraphs or semigroups, we will construct (under a small
additional geometric assumption) a boundary based on quasi-geodesic rays and
anti-rays that is preserved by quasi-isometries and, in the case of locally
finite digraphs and right cancellative semigroups, refines their ends. Among
other results, we show that it is possible to equip the space, if it is
finitely based, together with its boundary with a pseudo-semimetric.
