Prechains and self duality

We call prechain any binary relation ( V , ≺ ) for which the circular closure of the ternary relation x 1 → x 2 → x 3 is a circular ordering, where x → y means x ≺ y ⊀ x  ; i.e.  ( V , ≺ ) is a prechain if and only if there exists a linear strict ordering < on V such that for any x 1 , x 2 and x 3 in V , ( x 1 → x 2 → x 3 or x 2 → x 3 → x 1 or x 3 → x 1 → x 2 ) is equivalent to ( x 1 < x 2 < x 3 or x 2 < x 3 < x 1 or x 3 < x 1 < x 2 ). Thus a chain ( V , < ) , i.e. a set V endowed with a linear strict ordering < , is trivially a prechain. We characterize the class of prechains by a finite list of finite forbidden induced subrelations, and we give a description of those prechains. As an application, we obtain, for each integer k ≥ 4 , a description of the (possibly infinite) ( ≤ k ) -self dual binary relations. A binary relation is said to be ( ≤ k ) -self dual if each relation induced on at most k vertices is isomorphic to the relation obtained by reversing its arcs. That extends results previously known in the finite case, of which the proofs were obtained as byproducts of the description of difference classes w.r.t.  ( ≤ k ) -hypomorphy in reconstruction.