On Algebras with many symmetric operations

TL;DR

This paper characterizes finite algebras based on the presence of symmetric operations or specific automorphism properties, revealing a dichotomy in their structural symmetries.

Contribution

It establishes a new dichotomy linking symmetric term operations to automorphism properties in finite algebras.

Findings

Finite algebra either has symmetric operations of all arities or related automorphisms without common fixed points.

The two-automorphism condition cannot be simplified to a single fixed-point-free automorphism.

Provides a structural criterion connecting symmetry and automorphisms in algebraic varieties.

Abstract

We show that, for each finite algebra A, either it has symmetric term operations of all arities or else some finite algebra in the variety generated by A has two automorphisms without a common fixed point. We also show this two-automorphism condition cannot be replaced by a single fixed-point- free automorphism.