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A recursive approach for the determination of the nice sections of width three having a 4-crown stack as retract

Frank a Campo

TL;DR

The paper tackles the problem of characterizing finite minimal automorphic posets of width $3$ by focusing on nice sections and, in particular, those with a $4$-crown stack as retract. It introduces a recursive framework based on retractive down-splits and up-splits to detect when a crowned section admits a $4$-crown retract, and specializes to horizon-two nice sections $\mathfrak{N}_2$ to make the problem tractable. A structural isomorphism result shows that, for each height $n\ge2$, there are $2^{n-2}$ isomorphism types of posets in $\mathfrak{N}_2$, with an explicit construction and a standardized depiction of level-pair types. The authors develop a comprehensive procedure to enumerate and verify which posets in $\mathfrak{N}_2$ (up to height $6$) admit a $4$-crown retract, producing concrete examples and classifications that advance the understanding of minimal automorphic width-$3$ posets and illustrating the power of the recursive approach in a combinatorial setting.

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. In our article, we characterize those of them which have a 4-crown stack as retract and we develop a recursive approach for their determination. We apply the approach on a sub-class $\mathfrak{N}_2$ of nice sections of width three and determine all posets in $\mathfrak{N}_2$ with height up to six having a 4-crown stack as retract. For each integer $n \geq 2$, the class $\mathfrak{N}_2$ contains $2^{n-2}$ different isomorphism types of posets of height $n$.

A recursive approach for the determination of the nice sections of width three having a 4-crown stack as retract

TL;DR

The paper tackles the problem of characterizing finite minimal automorphic posets of width by focusing on nice sections and, in particular, those with a -crown stack as retract. It introduces a recursive framework based on retractive down-splits and up-splits to detect when a crowned section admits a -crown retract, and specializes to horizon-two nice sections to make the problem tractable. A structural isomorphism result shows that, for each height , there are isomorphism types of posets in , with an explicit construction and a standardized depiction of level-pair types. The authors develop a comprehensive procedure to enumerate and verify which posets in (up to height ) admit a -crown retract, producing concrete examples and classifications that advance the understanding of minimal automorphic width- posets and illustrating the power of the recursive approach in a combinatorial setting.

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. In our article, we characterize those of them which have a 4-crown stack as retract and we develop a recursive approach for their determination. We apply the approach on a sub-class of nice sections of width three and determine all posets in with height up to six having a 4-crown stack as retract. For each integer , the class contains different isomorphism types of posets of height .

Paper Structure

This paper contains 11 sections, 13 theorems, 17 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preparation
  3. Nice sections and crowned sections
  4. Definition and basic properties
  5. Four-crown stacks as retracts of crowned sections
  6. A recursive approach
  7. Nice sections with horizon two
  8. Definition and isomorphism types
  9. Properties of retracts
  10. Retractive up- and down-splits
  11. Application

Key Result

Lemma 2.1

Let $P$ be be a finite poset and $r : P \rightarrow R$ a retraction. There exists a retraction $v : P \rightarrow V$ with $V \simeq R$, $\min V \subseteq \min P$ and $\max V \subseteq \max P$.

Figures (7)

  • Figure 1: The level pairs $P(k,\ell)$, $k < \ell$, which are possible in an automorphic poset $P$ of width at most three.
  • Figure 2: Two nice sections with 4-crown stacks as retracts (hollow dots). The retraction of the 6-crown stack on the left is from Farley_1997.
  • Figure 3: The construction in the proof of Corollary \ref{['coro_P01_3C']}.
  • Figure 4: Illustrations for the proof of Corollary \ref{['coro_stackMitLuecke']}. The three levels show $P(k-1 \rightarrow k+1)$ for choices $b \in P(k-1)$, $w \in P(k+1)$ (which are not mandatory). The point $z$ is not mapped to $a$ by the retraction $s$ of $P(0 \rightarrow k-1)$.
  • Figure 5: a) The lower segment $10$ and a retraction $r$ onto a 4-crown with $r^{-1}(v) = \{ v \}$ for a point $v$ from the top level of the 4-crown. b) Illustrations for the case discrimination in the proof of Criterion \ref{['crit_split']}.5. The hollow dots belong to $S(0)$.
  • ...and 2 more figures

Theorems & Definitions (27)

  • Lemma 2.1: Duffus_etal_1980_DPR
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Theorem 3.4
  • Lemma 3.5
  • Corollary 3.6
  • proof
  • Lemma 3.7
  • proof
  • ...and 17 more