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Finite Simple Groups in the Primitive Positive Constructability Poset

Sebastian Meyer, Florian Starke

TL;DR

This paper connects the pp-constructability poset for finite structures to the algebraic structure of polymorphism clones, proving that any finite-clone with a quasi Maltsev operation and fully symmetric operations of all arities admits a minion homomorphism from the idempotent two-element clone $I$, thereby implying pp-constructability from $\mathbb{P}_1$. It then identifies the third layer's lower covers of $\mathbb{P}_1$ as the transitive tournament $\mathbb{T}_3$ and the family of structures $\mathbb{S}(G\curvearrowright\mathbb{P}(G))$ arising from finite simple groups, with explicit analyses for $A_5$, $\mathrm{PSL}(2,7)$, and $A_6$. The results show these lower covers are pairwise incomparable under pp-constructability and illuminate how finite simple group actions govern the lattice’s structure, including implications for descriptive complexity (e.g., Datalog fragments and linear programming). Overall, the work bridges finite group theory and the pp-constructability poset, highlighting the role of simple groups in the hierarchy of pp-constructable structures and providing a framework to study minimal blockers via group actions.

Abstract

We show that any clone over a finite domain that has a quasi Maltsev operation and fully symmetric operations of all arities has an incoming minion homomorphism from I, the clone of all idempotent operations on a two element set. We use this result to show that in the pp-constructability poset the lower covers of the structure with all relations that are invariant under I are the transitive tournament on three vertices and structures in one-to-one correspondence with all finite simple groups.

Finite Simple Groups in the Primitive Positive Constructability Poset

TL;DR

This paper connects the pp-constructability poset for finite structures to the algebraic structure of polymorphism clones, proving that any finite-clone with a quasi Maltsev operation and fully symmetric operations of all arities admits a minion homomorphism from the idempotent two-element clone , thereby implying pp-constructability from . It then identifies the third layer's lower covers of as the transitive tournament and the family of structures arising from finite simple groups, with explicit analyses for , , and . The results show these lower covers are pairwise incomparable under pp-constructability and illuminate how finite simple group actions govern the lattice’s structure, including implications for descriptive complexity (e.g., Datalog fragments and linear programming). Overall, the work bridges finite group theory and the pp-constructability poset, highlighting the role of simple groups in the hierarchy of pp-constructable structures and providing a framework to study minimal blockers via group actions.

Abstract

We show that any clone over a finite domain that has a quasi Maltsev operation and fully symmetric operations of all arities has an incoming minion homomorphism from I, the clone of all idempotent operations on a two element set. We use this result to show that in the pp-constructability poset the lower covers of the structure with all relations that are invariant under I are the transitive tournament on three vertices and structures in one-to-one correspondence with all finite simple groups.
Paper Structure (17 sections, 33 theorems, 40 equations, 4 figures)

This paper contains 17 sections, 33 theorems, 40 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Group actions
  4. Biactions
  5. Structures
  6. Clones
  7. Minor Conditions
  8. Primitive Positive Constructions
  9. The Submaximal Element
  10. The minor conditions satisfied by the clone of all idempotent operations
  11. Finite simple group actions
  12. Incomparable group actions
  13. Examples of Non-Abelian Simple Group Actions
  14. The Alternating Group of Degree Five
  15. The Projective Special Linear Group of Degree Two Over The Seven Element Field
  16. ...and 2 more sections

Key Result

Lemma 2.4

Let $G\curvearrowright X$ and $H\curvearrowright Y$ be group actions and let $t\in X^Y$. Let $Z_t=G(t)\cap H(t)\subseteq X^Y$. Then, the maps are bijections. Moreover, $\operatorname{Stab}_G(t)\trianglelefteq \operatorname{Stab}_G(H(t))\le G$ and $\operatorname{Stab}_H(t)\trianglelefteq \operatorname{Stab}_H(G(t))\le H$ and the above maps induce an isomorphism of groups.

Figures (4)

  • Figure 1: Some important structures from this article.
  • Figure 2: The $\{f,g\}$-reduct of ${\mathbb S}(A_5\curvearrowright\mathbb{P}(A_5))$, where the relation for $g$ is represented with undirected dashed edges.
  • Figure 3: The $\{f,g\}$-reduct of ${\mathbb S}(\operatorname{PSL}(2,7)\curvearrowright\mathbb{P}(\operatorname{PSL}(2,7)))$, where the relation for $g$ is represented with undirected dashed edges.
  • Figure 4: The $\{f,g\}$-reduct of ${\mathbb S}(A_6\curvearrowright\mathbb{P}(A_6))$, where the relation for $g$ is represented with undirected dashed edges.

Theorems & Definitions (63)

  • Definition 2.1
  • Lemma 2.4
  • proof
  • Definition 2.5
  • Definition 2.6
  • Theorem 2.7: Theorem 1.3 in wonderland
  • Theorem 2.8: Theorem 3.12 in vucaj2024submaximal and Theorem 6.1.13 in AlbertThesis
  • Lemma 2.9: Lemma 3.4 in vucaj2024submaximal and Lemma 6.1.5 in AlbertThesis
  • Theorem 3.1
  • Definition 3.2
  • ...and 53 more