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

Appendix

Fig. 7

Admissible extensions R ∈ E(P, m) with P ∈ Z(k) (cf. Table 1)

Fig. 8

Admissible extensions R ∈ E(C₂ + A_ℓ,m), ℓ ≥ 1 (cf. Table 2). The coding is: type of C₂ in R, a = max{ϕ(R) (y) | y ∈ Y}

Fig. 9

Admissible extensions R ∈ E(P′ + A_ℓ, m), P′ ∈ Z(k′), k′ ≥ 3, ℓ ≥ 1 (cf. Table 3)

Fig. 10

Admissible extensions R ∈ E(A_k,m) with k ≤ m (cf. Table 4). The admissible extensions with k > m are found by turning upside-down the admissible extensions found in E(A_m,k)

Fig. 11

Non-connected lower extensions R without isolated points with $\delta (R) \geq \frac {1}{4}$ (cf. Table 5). In order to be consistent with the previous figures, m is in this figure for every extension the number of minimal points of the connectivity component on the right. Each connectivity component can be replaced by its dual

Table 1 The binary representation of the factors of the admissible extensions R ∈ (P, m) with P ∈ Z(k) (Fig. 7) and the duals of these extensions

Table 2 The binary representation of the factors of the admissible extensions contained in (C₂ + A_ℓ,m),ℓ ≥  1 (Fig. 8) and the duals of these extensions

Table 3 The binary representation of the factors of the admissible extensions contained in 𝓔(P′ + A_ℓ, m), P′ ∈ Z(k′), k′ ≥ 3, ℓ ≥ 3 (Fig. 9) and the duals of these extensions

Table 4 The binary representation of the factors of the admissible extensions contained in (A_k,m) with k ≤ m (Fig. 10)

Table 5 The binary representation of the factors of the non-connected lower extensions R without isolated points with δ(R) ≥ 1 4 (Fig. 11)