Quasi-isometry invariance of hyperbolicity in semimetric spaces, digraphs and semigroups

Gray and Kambites introduced a notion of hyperbolicity in the setting of semimetric spaces like digraphs or semigroups. We will prove that under a small additional geometric assumption their notion of hyperbolicity is preserved by quasi-isometries. Applied to semigroups, this will partially solve a problem of Gray and Kambites.
Comment: 16 pages, previous version (v1) is split into two parts, this is the first part