Finite Simple Groups in the Primitive Positive Constructability Poset

TL;DR

This paper explores the structure of clones with specific algebraic operations over finite domains, revealing connections to finite simple groups and characterizing minimal elements in the pp-constructability poset.

Contribution

It establishes a link between clones with quasi Maltsev and symmetric operations and finite simple groups within the pp-constructability framework.

Findings

Identifies the minimal structures in the pp-constructability poset related to finite simple groups.

Shows that certain clones have a homomorphism from the clone of all idempotent operations on a two-element set.

Characterizes the lower covers of structures invariant under all operations in a specific clone.

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.