The conjecture on p.467 of the paper has been proved :

Robert Bieri and J.R.J.Groves,
*The geometry of the set of characters induced by valuations*,
J. reine u. angew. Math. **347** (1984) 168-195.
MR **86c**:14001.

Cf. also

Robert Bieri and J.R.J.Groves,
*A rigidity principle for the set of all characters induced by
valuations,*
T.A.M.S. **294** (1986) 425-434.
MR **87i**:16015.

The results of the paper itself have been used by group theorists, e.g.,

J. E. Roseblade, *Group rings of polycyclic groups*
J. Pure Appl. Algebra **3** (1973) 307-328.
MR **48**#11269.

Another paper extending the ideas of my paper is

Daniel R. Farkas,
*The Diophantine nature of some constructions at infinity*,
Geometry of group representations (Boulder, CO, 1987), 125-129,
Contemp. Math., **74**. MR **89m**:20008.

See also the book

Bernd Sturmfels,
*Solving systems of polynomial equations,*
CBMS Regional Conference Series in Mathematics, v.97,
AMS, 2002. viii+152 pp.
MR**2003i**:13037

ERRATA:

The second half of section 3 concerns certain families
of subgroups *G* of **Z**^{n} which arise as
kernels of homomorphisms to ordered groups.
Clearly, every such subgroup is *pure*, equivalently
(since **Z**^{n} is torsion-free),
satisfies *nx* ∈ *G* ⇒
*x* ∈ *G* for positive integers *n*.
With that case
in mind, I unfortunately made some general assertions about families
of subgroups of **Z**^{n} that are not true
without a purity hypothesis.
In particular, the statement on p.462, end of paragraph
slightly above the center, "for *T* > *S* we
have (*p*_{1}(*T*), ...,
*p _{n}*(

There are several ways these problems can be fixed.
The simplest is
to define a "subgroup system" to be a finite nonempty family of
*pure* subgroups of **Z**^{n}.
Then the above incorrect assertions become correct.
The one modification this entails is in the description
of *S* ∨ *T*, which will then consist, not of the
sums *G* + *H*
with *G* ∈ *S*,
*H* ∈ *T*, but of the
division-closures of those sums in **Z**^{n}.

A similar solution is to replace the concept of a "subgroup system" by
that of a "subspace system", meaning a finite family of vector subspaces
of **Q**^{n}.
The general statements about subgroup systems in
section 3 become correct for these subspace systems, but a little
more work is needed in connecting these results with our hypotheses
and conclusions: We must note how to obtain from our homomorphisms
on **Z**^{n} subspaces
of **Q**^{n}, and then having
concluded that that the stabilizer
of *I* in GL(*n*, **Z**)
stabilizes a nontrivial
subspace system *S*, we must note that it
therefore stabilizes a nontrivial finite family of subgroups
of **Z**^{n},
namely {*G* ∩ **Z**^{n}
| *G* ∈ *S*}.

As a third alternative, one could keep the original definition of a
"subgroup system"
(though it is a finer tool than required), and modify our arguments.
Despite the incorrectness of the proof using lexicographic order,
it is true that the lattice of all subgroup systems of
**Z**^{n} has ascending chain condition; this
is a case of a general lemma proved at the end of this errata-page.
And the fact, needed at the end of section 3, that
*Q* consists of subgroups
of infinite index in **Z**^{n} can
be seen by noting
that if it did not, then on dropping all subgroups of finite index, one
would have a subgroup system majorizing *Q* under our
partial ordering and still compatible with *V*.

I am indebted to Greg Marks for pointing out the errors corrected above, and a couple of those noted in the following list of minor corrections:

P.461, last line of first full paragraph:
*X*^{α} should be
*x*^{α}.

P.463, top line: After "other elements" add "*t*", and at the end
of the line, change "relation" to "relations".

P.464, first sentence of proof of Theorem 2: In (2), &supe should be &sube, and on the next line, after "finally", add "(4)".

P.465, line after first display: In the first inequality, both
occurrences of max should be max^{(2)}
and inversely in the second inequality, both occurrences
of max^{(2)} should be max.

(To see the former inequality, consider two cases:
If |*c*_{α} *x*^{α}| is
maximized by the same α that
maximizes |*x*^{α}|, then all other
terms |*c*_{α} *x*^{α}|
involve an |*x*^{α}| term that
is ≤ max^{(2)}(|*x*^{α}|), and
the result clearly holds.
In the contrary case, the maximum value
of |*c*_{α} *x*^{α}| is
itself ≤
*M* max^{(2)}(|*x*^{α}|), hence
a fortiori so is the max^{(2)} value.
The second inequality is straightforward.)

Throughout the paper, I often wrote "designate" where I would now use "denote".

LEMMA REFERRED TO UNDER "ERRATA" ABOVE:

The fact that the set of subgroup
systems **Z**^{n} has ACC
is a special case of the following lemma, probably
known to people in the field of partially ordered sets.
I would be grateful to anyone who could give me a reference.
(Note that the antichains of the lemma correspond to the
*irredundant* subgroup systems of the paper.)

**Lemma**.
Let *P* be any partially ordered set with ACC,
and let FA(*P*) be the partially ordered set of finite
antichains in *P* (finite subsets with no comparable
elements), ordered by making *T* ≥ *S* if every
member of *T* is ≥ some member
of *S*.
Then FA(*P*) also has ACC.

*Proof*.
Let *S*_{0} ≤
*S*_{1} ≤ ... in
FA(*P*); it will clearly suffice to show that
∪_{i}* S _{i}* is finite.
Given

Assuming the contrary, let us choose *m* ≥ 0
and *s _{m}* ∈

But note that any sequence *s _{m}*
≤

*Afterthought:* If we associate to every finite
antichain *S* in *P* the dual ideal
(subset of *P* closed under passing to larger
elements) that it generates, we get a bijection between the
set of finite antichains and the set of finitely generated dual
ideals of *P*.
The ordering described above then corresponds
to the opposite of the inclusion ordering on dual ideals.
Hence, reversing all orderings, the above lemma translates to
say that if *P* has DCC, then the set of its finitely
generated ideals,
partially ordered by inclusion, also has DCC.
It seems most likely to
be known in something like this form to people who study
ordered sets.