(1) Follows by definition of M[U]and N[U].

(2) This is a direct consequence of (1).

(3) Suppose M ⊆ N and M[U] ( N[U]. Then there must be some J ∈ N such that J
[U] ∈/ M[U]. Consequently, J /∈ M and so M must be a proper subset of N.

## A.2 Saturated Mknf Interpretations

An interesting class of MKNF interpretations are *saturated* MKNF interpretations. As we will see, all MKNF models of some formula or set of formulae are saturated in a certain sense. Furthermore, a strengthened version of Proposition 27 can be shown for saturated MKNF interpretations, with implications replaced by equivalences. In this subsection we formally define the class of saturated MKNF interpretations and then we prove some of their properties.

Definition 28 (*Saturated MKNF Interpretation*)
Let U be a set of predicate symbols and M ∈ M. We say an M is *saturated relative to* U
if for every interpretation I ∈ I the following holds:
If I
[U] ∈ M[U], then I ∈ M.

Interestingly, all MKNF models of a theory T are saturated relative to the set of predicate symbols relevant to T :
Proposition 29 Let U be a set of predicate symbols, T be an MKNF theory such that U ⊇ pr(T ) and M
be an MKNF model of T . Then M is saturated relative to U.

Proof Suppose M is not saturated relative to U. Then there is some I ∈ I such that I
[U] ∈
M[U]and I /∈ M. Let M′ = M ∪ { I }. M is an MKNF model of T , so by definition hI
′, M′, M**i 6|**= T for some I
′ ∈ M′. But I
′[U] ∈ M′[U] = M[U], so there must be some I
′′ ∈ M such that I
′′[U] = I
′[U]. By two applications of Proposition 25 we now obtain hI
′, M′, M**i 6|**= T =⇒
DI
′[U], M′[U], M[U]E6|= T =⇒ hI
′′, M, M**i 6|**= T .

This is in conflict with the assumption that M is an MKNF model of T .

Proposition 30 Let U be a set of predicate symbols and M, N ∈ M be such that M saturated relative to U. Then the following equivalences hold:
(1) M = N if and only if M ⊆ N and M[U] ⊇ N[U]. (2) M ( N if and only if M ⊆ N and M[U] ( N[U].

Proof