There exists a nonrecursive c.e. set A such that for every X ≡_m A there is a c.e. B ≡_m A with X ≰_fo B, so the m-degree of A contains no least finite-one degree.
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A computably enumerable many-one degree with no least finite-one degree
There exists a nonrecursive c.e. set A such that for every X ≡_m A there is a c.e. B ≡_m A with X ≰_fo B, so the m-degree of A contains no least finite-one degree.