
204
Przymusinski
PROPOSITION 4
The predicate priority relation <
p
is noetherian iff it is a partial order.
Moreover, if <
p
is noetherian then so is <.
Proof:
If there exists an infinite increasing (in the sense of <
p
) sequence of
elements of P, then two of its elements must have the same predicate symbol
A, which implies that A <
p
A and shows that <
p
is not irreflexive. The con-
verse is obvious.
If there exists an infinite increasing (in the sense of <) sequence of ele-
ments of H then two of its elements must have the same predicate symbol,
which implies that <
p
is not noetherian. •
The property that the relation < (resp. the relation <
p
)