The purpose of this article is to introduce the theory of presentations of S-posets. We construct general presentations for various S-poset constructions related to sub S-posets and Rees quotients. More precisely, given an S-poset A and a sub S-poset B of A, on the one hand, we construct presentations for B and the Rees quotient A=B using a presentation for A, and on the other hand, we derive a presentation for A from presentations for B and A=B. We also construct a general presentation for the union of two sub S-posets. From our general presentations, we deduce a number of finite presentability results. Finally, we consider the case where a sub S-posets B has a finite complement in an S-posets A. We show that if S is a finitely generated pomonoid and B is finitely presented, then A is finitely presented. We also show that if S belongs to a wide class of all finitely presented pomonoids then the converse also holds.
