Preservation of almost transitivity under additive closure

Determine whether, for a relation R on a W-semigroup (S,≺) that is ≺-almost transitive, the additive closure R+ is also ≺-almost transitive; equivalently, establish or refute that ≺◦R◦≺◦R◦≺ ⇒ ≺◦R◦≺ implies ≺◦R+◦≺◦R+◦≺ ⇒ ≺◦R+◦≺.

Background

Section 7 studies generating normal pairs from relations and the effect of additive structure on transitivity properties. The authors introduce the notion of ≺-almost transitivity and highlight uncertainty about whether this property survives taking additive closures, which are central in defining quotient structures from relations.

They provide sufficient conditions via almost refinement (Proposition 7.10) under which the preservation holds, but the general status remains unresolved, motivating a precise question about additive closure and almost transitivity.