Let {We}e∈ω be a standard enumeration of the recursively enumerable (r.e.) subsets of ω = {0,1,2,…}. The lattice of recursively enumerable sets, , is the structure ({We}e∈ω,∪,∩). ≡ is a congruence relation on if ≡ is an equivalence relation on and if for all U, U′ ∈ and V, V′ ∈ , if U ≡ U′ and V ≡ V′, then U ∪ V ≡ U′ ∪ V′ and U ∩ V ≡ U′ ∩ V′. [U] = {V ∈ | V ≡ U} is the equivalence class of U. If ≡ is a congruence relation on , the elements of the quotient lattice / ≡ are the equivalence classes of ≡. [U] ∪ [V] is defined as [U ∪ V], and [U] ∩ [V] is defined as [U ∩ V]. We say that a congruence relation ≡ on is if {(i, j)| Wi ≡ Wj} is . Define =* by putting Wi, =* Wj if and only if (Wi − Wj)∪ (Wj − Wi) is finite. Then =* is a congruence relation. If D is any set, then we can define a congruence relation by putting Wi Wj if and only if Wi ∩ D =* Wj ∩D. By Hammond [2], a congruence relation ≡ ⊇ =* is if and only if ≡ is equal to for some set D.

The Friedberg splitting theorem [1] asserts that if A is any recursively enumerable set, then there exist disjoint recursively enumerable sets A0 and A1 such that A = A0∪ A1 and such that for any recursively enumerable set B