On Morita Equivalence of Partially Ordered Monoids

We show that there is one-to-one correspondence between certain algebraically and categorically defined subobjects, congruences and admissible preorders of S-posets. Using preservation properties of Pos-equivalence functors between Pos-categories we deduce that if S and T are Morita equivalent partially ordered monoids and F:Pos _S →Pos _T is a Pos-equivalence functor then an S-poset A _S and the T-poset F(A _S ) have isomorphic lattices of (regular, downwards closed) subobjects, congruences and admissible preorders. We also prove that if A _S has some flatness property then F(A _S ) has the same property.