On zigzag theorem for commutative pomonoids and certain closed and absolutely closed monoids and pomonoids

The Isbell Zigzag Theorem for pomonoids was first proved by Sohail Nasir in 2013. In this paper we give a short proof of the Zigzag Theorem for commutative pomonoids. We also prove that certain submonoids of full transformation monoid on a non empty set are closed in the containing monoids. In particular we prove that the monoid of all surjective mappings and the pomonoid of all monotone mappings are absolutely closed.