$\mathcal{G}$-systems

A $\mathcal{G}$-system is a collection of $\mathbb{Z}$-bases of ${\mathbb{Z}}^n$ with some extra axiomatic conditions. There are two kinds of actions “mutations” and “co-Bongartz completions” naturally acting on a $\mathcal G$-system,whichprovidethecombinatorialstructureofa$\mathcal G$-system. It turns out that “co-Bongartz completions” have good compatibility with “mutations”. The constructions of “mutations” are known before in different contexts, including cluster tilting theory, silting theory, $\tau$-tilting theory, cluster algebras, marked surfaces. We found that in addition to “mutations”, there exists another kind of actions “co-Bongartz completions” naturally appearing in these different theories. With the help of “co-Bongartz completions” some good combinatorial results can be easily obtained. In this paper, we give the constructions of “co-Bongartz completions” in different theories. Then we show that $\mathcal{G}$-systems naturally arise from these theories, and the “mutations” and “co-Bongartz completions” in different theories are compatible with those in $\mathcal{G}$-systems.