3 problems

Lozin's fourth-operation conjecture. The image of any $f_{n,k}$ under this operation again has minimum specification number, that is, belongs to $df_{n+2}$.

Lozin's fourth-operation conjecture. Applying this operation to any $f_{n,k}$ produces a member of $df_{n+2}$.

Nested-function characterization. The nested functions should be the only functions $f e?$ in $dan$ satisfying