In mathematics, Markowsky's theorem states: every chain-complete poset is a dcpo where
a poset is chain-complete if each chain in it has a least upper bound. a poset is a dcpo if each directed set in it has a least upper bound. Since a dcpo is chain-complete (as a chain is directed), the converse of the theorem is trivial. A known proof uses Iwamura's lemma and ordinals.
References
Markowsky, George (1976). "Chain-complete posets and directed sets with applications". Algebra Universalis. 6: 53–68. doi:10.1007/BF02485815. Goubault-Larrecq, Jean (February 23, 2015). "Iwamura's Lemma, Markowsky's Theorem and ordinals". Retrieved January 6, 2024. Goubault-Larrecq, Jean (January 28, 2018). "Markowsky or Cohn?". Retrieved January 6, 2024.