The notion of pseudoorder on a partially ordered (p.o.) semigroup \((S, \cdot, \leq)\) was introduced by the authors in an earlier paper [ibid. 50, No. 2, 161-177 (1995; Zbl 0823.06010)]. A transitive, left- and right- compatible relation \(\rho\) on \(S\) is a pseudoorder if \(a \leq b\) implies \(a \rho b\) (consequently, \(\rho\) is reflexive). Notice that for every homomorphism \(\varphi : (S, \cdot, \leq) \to(T, *, \preceq)\) of p.o. semigroups the relation \(\rho\) on \(S\) defined by: \(a \rho b \Leftrightarrow \varphi (a) \preceq \varphi (b)\), is a pseudoorder. It allows to define for the congruence \(\overline \rho = \rho \cap \rho^{- 1}\) on \(S\) a partial ordering on \(S/ \overline \rho\) which makes \((S/ \overline \rho, *, \preceq)\) a p.o. semigroup: \(a \rho \preceq b \rho \Leftrightarrow a \rho b\). This is the reason for replacing arbitrary congruences on \((S, \cdot)\) by congruences \(\overline \rho\) containing th partial order \(\leq\) of \(S\) \((\preceq\) on \(S\overline \rho\) is in a certain sense induced by \(\leq\) on \(S)\). Using the concept of pseudoorder the well-known homomorphism and isomorphism theorems for semigroups \((S,\cdot)\) are extended to p.o. semigroups \((S, \cdot, \leq)\).