Abstract
Let P and P' be two orders on the same set X. The order P' is hemimorphic to P if it is isomorphic to P or to its dual P star. It is (<= 4)hemimorphic, respectively hereditarily hemimorphic, to P if for each subset A of X with broken vertical bar A broken vertical bar <= 4, respectively for each subset A of X, the orders PIA and P(A induced on A are hemimorphic. In this paper, we begin with proving that, given a connected order P, if an order P' is (<= 4)-hemimorphic to P, then either P'((sic)A) and P-(sic)A are isomorphic for each finite subset A of X or P-(sic)A(') and P-(sic)A* are isomorphic for each finite rA rA subset A of X. Then we show that no module of an order P is an infinite chain and at most one connected component of P is not self dual, respectively is not a chain, if and only if P' is hemimorphic, respectively hereditarily hemimorphic, to P whenever P' is any order (<= 4)hemimorphic to P.