Upper Covers of Chains and Antichains in Sets of Indecomposable Subsets

We prove that there are arbitrarily large indecomposable ordered sets T with a 2-chain C such that the smallest indecomposable proper superset U of C in T is T itself. Subsequently, we characterize all such indecomposable ordered sets T and 2-chains C. We also prove the same type of result for 2-antichains.