Lattice properties of Rogers semilattices of compuatble and generalized computable families

摘要

We consider the distributivity property and the property ofbeing a lattice of Rogers semilattices of generalized computable families.We prove that the Rogers semilattice of any nontrivial A-computablefamily is not a lattice for every non-computable set A. It is also provedthat if a set A is non-computable then the Rogers semilattice of anyinfinite A-computable family is not weakly distribuive. Furtermore, wefind two infinite computable families with nontrivial distributive andproperly weakly distributive nontrivial Rogers semilattices.