A new kind of anti-foundation axioms

Among the four well-known anti-foundation axioms, $\mathsf{BAFA}$, $\mathsf{FAFA}$, $\mathsf{SAFA}$, and $\mathsf{AFA}$, the latter three are all special cases of $\mathsf{AFA}^\sim$ ($\sim$ is one of the regular bisimulations). In this paper, we generalize it to $\mathsf{AFA}^\sim$ ($\sim$ is regular and has the so-called set property), and show that this generalization is substantial by constructing new anti-foundation axioms.