Negative numberings in admissible sets. I

We construct an admissible set $\mathbb{A}$ such that the family of all $\mathbb{A}$-computably enumerable sets possesses a negative computable $\mathbb{A}$-numbering but lacks positive computable $\mathbb{A}$-numberings. We also discuss the question on existence of minimal negative $\mathbb{A}$-numberings.