A contribution to the characterization of finite minimal automorphic posets of width three

TL;DR

The work tackles the open problem of characterizing finite minimal automorphic posets of width three by narrowing to the sub-class $\mathfrak{N}_2$ of horizon-two nice sections and providing a constructive, recursive framework. It offers a precise structural characterization for non-trivial width-three retracts (towers of nice sections) and for retracts that are 4-crown stacks, together with a method to build and recognize such retracts via retractive up-splits and down-splits. Leveraging these results, the authors enumerate all posets in $\mathfrak{N}_2$ with height up to six that admit a 4-crown retract and establish that for each height $n\ge 2$, there are exactly $2^{n-2}$ isomorphism types, yielding concrete classifications and practical tools for fixed-point property analyses in width-three automorphic posets.

Abstract

The characterization of the finite minimal automorphic posets of width three is still an open problem. Niederle has shown that this task can be reduced to the characterization of the nice sections of width three having a non-trivial tower of nice sections as retract. We solve this problem for a sub-class $\mathfrak{N}_2$ of the finite nice sections of width three. On the one hand, we characterize the posets in $\mathfrak{N}_2$ having a retract of width three being a non-trivial tower of nice sections, and on the other hand we characterize the posets in $\mathfrak{N}_2$ having a 4-crown stack as retract. The latter result yields a recursive approach for the determination of posets in $\mathfrak{N}_2$ having a 4-crown stack as retract. With this approach, we determine all posets in $\mathfrak{N}_2$ with height up to six having such a retract. For each integer $n \geq 2$, the class $\mathfrak{N}_2$ contains $2^{n-2}$ different isomorphism types of posets of height $n$.

Figures (10)

• Figure 1: Posets of type $22, 32, 23, 33, 2C$, and $3C$.
• Figure 2: A nice section $P$ of width three with two retracts $R$ and $T$. The arrows indicate a retraction onto $R$. $R$ is not a tower of sections but $T$ is a tower of nice sections.
• Figure 3: Two nice sections with 4-crown stacks as retracts.
• Figure 4: Examples for the retractions described in Lemma \ref{['lemma_R0P0_C3']}.
• Figure 5: A poset $P \in \mathfrak{ N }_2$ having a non-trivial retract $R$ of width three being a tower of nice sections. The segments $U$, $V$, and $W$ refer to Theorem \ref{['theo_32']}.

Theorems & Definitions (51)

• Lemma 2.1
• proof
• Lemma 2.2: Schroeder_2016
• Definition 3.1
• Lemma 3.2: Niederle_1989
• Proposition 3.3
• proof
• Corollary 3.4
• proof
• Definition 3.5