
600
Lassez,
Mäher,
and
Marriott
LEMMA
3
Any nonempty finite or infinite collection of solvable equation sets has only a
finite number of generalizations, modulo
ä
.
Proof:
Let the collection be {E
x
,...,
£„,...}.
Let [g-] be the set of distinct
generalizations of
{£j,...,
£,·}, modulo ~. Clearly [g
x
] D [g
2
] 2 ··· 2 [g
n
] 2
.... From the above lemma there are only a finite number of elements in [g
x
]
and so this lemma follows. •
As a consequence of this lemma, the poset of equation sets modulo « has
no infinite, strictly increasing sequence.
THEOREM 5
Any nonempty, possibly infinite, collection of solvable equation sets has an
msg which is uniqu ...