Upper Covers of Chains and Antichains in Sets of Indecomposable Subsets

TL;DR

This paper investigates the structure of large indecomposable ordered sets containing specific 2-chains or 2-antichains, characterizing when such sets are minimal supersets, advancing understanding of their combinatorial properties.

Contribution

It provides a characterization of indecomposable ordered sets containing particular 2-chains and 2-antichains, revealing conditions for minimal supersets in these structures.

Findings

Existence of arbitrarily large indecomposable sets with specific 2-chains

Characterization of all such indecomposable sets and chains

Extension of results to 2-antichains

Abstract

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.