The submonoid and rational subset membership problems for Artin groups

We demonstrate that the submonoid membership problem and the rational subset membership problem are equivalent in Artin groups. Both problems are undecidable in a given Artin group if and only if the group embeds the right-angled Artin groups of rank 4 over a path or a square; and this can be characterized using only the defining graph of the Artin group. These results generalize those of Lohrey - Steinberg for right-angled Artin groups. Moreover, both decision problems are decidable for a given Artin group if and only if the group is subgroup separable.