A Framework for the Systematic Determination of the Posets on n Points with at Least τ ⋅ 2 n Downsets

Abstract

A structural framework is presented which allows the systematic determination of all non-isomorphic posets on n points with at least τ ⋅ 2n downsets, or - equivalently - of all T0-topologies on n points with at least τ ⋅ 2n open sets. The framework is developed by defining the type of a special extension of posets and by defining a partial order on these types. It is shown that this partial order is strictly antitone to the number of downsets of so called simple extensions; additionally the downset number of a simple extension is the maximum of the downset numbers of all extensions having the same type as the simple extension. For non-simple extensions of an important type it is shown that the number of their downsets is ruled by a simple sub-extension contained in them. The approach is used to determine all non-isomorphic posets with at least \\( \\frac {1}{4} \\cdot 2^{n}\\) downsets. Finally, a result of Vollert about the structure of downset numbers is refined.