Prechains and self duality

We call prechain   any binary relation (V,≺)(V,≺) for which the circular closure of the ternary relation x1→x2→x3x1→x2→x3 is a circular ordering, where x→yx→y means x≺y⊀xx≺y⊀x ; i.e.   (V,≺)(V,≺) is a prechain if and only if there exists a linear strict ordering << on VV such that for any x1x1, x2x2 and x3x3 in VV, (x1→x2→x3x1→x2→x3 or x2→x3→x1x2→x3→x1 or x3→x1→x2x3→x1→x2) is equivalent to (x1