Lattice properties of Rogers semilattices of compuatble and generalized computable familie

We consider the distributivity property and the property of being a lattice of Rogers semilattices of generalized computable families. We prove that the Rogers semilattice of any nontrivial $A$-computable family is not a lattice for every non-computable set $A$. It is also proved that if a set $A$ is non-computable then the Rogers semilattice of any infinite $A$-computable family is not weakly distribuive. Furtermore, we find two infinite computable families with nontrivial distributive and properly weakly distributive nontrivial Rogers semilattices.