The local-global property for G-invariant terms

TL;DR

This paper studies the local-global property for G-invariant terms in algebraic structures, showing that cyclic loop conditions have this property and can be decided efficiently, while symmetric terms of higher arity do not.

Contribution

It establishes the presence or absence of the local-global property for various G-invariant terms, impacting their computational decidability.

Findings

Cyclic loop conditions have the local-global property and are decidable in polynomial time.

Symmetric terms of arity greater than 2 lack the local-global property.

The results connect algebraic properties with computational complexity.

Abstract

For some Maltsev conditions $\Sigma$ it is enough to check if a finite algebra $\mathbf A$ satisfies $\Sigma$ locally on subsets of bounded size, in order to decide, whether $\mathbf A$ satisfies $\Sigma$ (globally). This local-global property is the main known source of tractability results for deciding Maltsev conditions. In this paper we investigate the local-global property for the existence of a $G$-term, i.e. an $n$-ary term that is invariant under permuting its variables according to a permutation group $G \leq$ Sym($n$). Our results imply in particular that all cyclic loop conditions (in the sense of Bodirsky, Starke, and Vucaj) have the local-global property (and thus can be decided in polynomial time), while symmetric terms of arity $n>2$ fail to have it.