Orders and (≤4)-Hemimorphy
Mohammad Alzohairi, Moncef Bouaziz and Youssef Boudabbous
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. It is (≤ 4)- hemimorphic, respectively hereditarily hemimorphic, to P if for each subset A of X with |A| ≤ 4, respectively for each subset A of X, the orders PA 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 A and PA are isomorphic for each finite subset A of X or P A and P A are isomorphic for each finite 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.