(Homo) Flatness of posets on poideal extensions

We consider pomonoids , where G is a pogroup and I is a poideal of S and show that if an S-poset is principally weakly flat, (weakly) flat, po-flat, (principally) weakly po-flat, (po-) torsion free or satisfies Conditions (P), (P _E ), (P _w ), (PWP), (PWP) _w , (WP) or (WP) _w as an I ¹-poset, then it has these properties as an S-poset. We also show that an S-poset which is free, projective or strongly flat as an I ¹-poset may not generally have these properties as an S-poset.     

Zigzag Theorem for Partially Ordered Monoids

While investigating the connection of absolute flatness of partially ordered monoids (briefly pomonoids) with their amalgamation, the author recently posed the question as to whether it was possible to formulate some ordered analogue of Isbell's zigzag theorem. Proving such a theorem is the primary aim of this article.