 Chain-complete partially ordered set

A chain-complete partially ordered set has a supremum for every chain, with the empty-chain condition requiring a least element if empty chains are included. This implies the weaker upper-bound hypothesis of Zorn lemma, which requires only that each chain have some upper bound, not necessarily a least one. Some treatments use “chain-complete” for that weaker condition, so the convention should be specified. In the inclusion order on lattice ideals avoiding a specified element, the union of a nonempty chain is an ideal and is its supremum. This union observation is the step needed to apply Zorn in prime-ideal separation arguments.