“…In some constructions it is useful to apply the following lemma (called also the Ends-lemma having many applications-cf. [25][26][27][28][29]). Recall, first that by a (quasi-)ordered semigroup we mean a triad (S, ·, ≤), where (S, ·) is a semigroup, (S, ≤) is a (quasi-)ordered set, i.e., a set S endowed with a reflexive and transitive binary relation "≤" and for all triads of elements a, b, c ∈ S the implication…”