For a poset \({\mathcal P}=(P,\leq)\) the associated PT-order is the reflexive and transitive binary relation \(\trianglelefteq\) in which \(a\trianglelefteq b\) holds if every maximal chain of \({\mathcal P}\) which passes through a also passes through b. A cutset of \({\mathcal P}\) is a subset of P which has a nonempty intersection with every maximal chain of \({\mathcal P}\). The poset \({\mathcal P}\) is called special if whenever A is a chain in \({\mathcal P}\) and \(a=\sup A\) or inf A, then there is \(b\in A\) such that \(b\trianglelefteq a\). Further \({\mathcal P}\) is said to be chain complete if every chain \(A\subseteq P\) has an infimum and a supremum in \({\mathcal P}\). Finally, \({\mathcal P}\) is said to be regular if it is chain complete and if whenever A is a chain in \({\mathcal P}\), \(a=\sup A\) (inf A) and \(xa)\), then there is \(b\in A\) such that \(xb)\). Deriving from more general theorems, the author proves the following facts: 1) If \({\mathcal P}\) is chain complete and special, then the set of \(\trianglelefteq\)-maximal elements is \(\trianglelefteq\)-dominating (i.e. for every \(y\in P\), there is a \(\trianglelefteq\)-maximal element x such that \(y\trianglelefteq x)\) and contains a minimal cutset. 2) If \({\mathcal P}\) is regular and special, then it is a union of minimal cutsets (partial answer to a question of Rival and Zaguia). 3) If \({\mathcal P}\) is chain complete and special, then min\(\{\) \(| X|:\) X is a cutset of \({\mathcal P}\}=\sup \{| M|:\) M is a set of pairwise disjoint maximal chains in \({\mathcal P}\}\) and the supremum is attained (partial answer to a question of Brochet and Pouzet).