On the rational subsets of the monogenic free inverse monoid

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We prove that the equality problem is decidable for rational subsets of the monogenic free inverse monoid F. It is also decidable whether or not a rational subset of F is recognizable. We prove that a submonoid of F is rational if and only if it is finitely generated. We also prove that the membership problem for rational subsets of a finite J-above monoid is decidable, covering the case of free inverse monoids.