arXiv:1801.06779v1 [] 21 Jan 2018

ON THE CONSTRUCTION OF ATOMIC AND
ANTIMATTER SEMIGROUP ALGEBRAS
FELIX GOTTI
Abstract. In this paper, we construct three families of semigroup algebras using
Puiseux monoids, recently-studied additive submonoids of Q. The first family consists
of semigroup algebras of certain atomic Puiseux monoids called primary. Properties
of primary Puiseux monoids will allow us to show that the semigroup algebras they
determine are atomic. The second family of algebras is constructed over a given
algebraically closed field by using integrally closed Puiseux monoids. As we shall
prove, the algebraically closedness of the field and the integrally closedness of the
monoids ensure that the semigroup algebras they determine are antimatter Bezout
domains. Then, we use a class of divisible Puiseux monoids to construct our last
family of antimatter semigroup algebras over any perfect field of finite characteristic.

1. Introduction
The study of group rings and algebras dates back to the mid-nineteen century. Most
of the initial research in this area focused mainly on the structure of the groups of
units of group rings and the isomorphic problem for groups over a given coefficient
ring; see Highman [19] and Sehgal [20], respectively. However, most of the early initial
ring-theoretical study of group rings and algebras was mainly carried out on a noncommutative setting.
It was not until the seventies that the study of a commutative version of group rings
started earning substantial attention. In a series of papers, Gilmer and Parker began a
systematic treatment of semigroup rings (see [10, 12, 13]). They particularly focused on
the following problem: given a commutative ring R and a commutative semigroup S,
establish conditions under which the semigroup ring R[X; S] of S over R satisfies certain
algebraic property. Clearly, the determination of such conditions often requires a fair
understanding of the algebraic properties of S and R. As the structure of commutative
semigroups most of the time cannot be derived from that one of abelian groups, a new
research direction in commutative algebra had emerged.
Since then semigroup rings and algebras have received a significant amount of consideration. Semigroup algebras, in particular, permeate through many fields under
Date: January 23, 2018.
Key words and phrases. Puiseux monoids, system of sets of lengths, sets of lengths, realization
theorem, characterization problem, atomic monoids.
1

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F. GOTTI

active research, including algebraic combinatorics [4], discrete geometry [5], and functional analysis [1]. Gilmer in [11] offers a comprehensive exposition on the advances of
commutative semigroup ring theory until mid 1980.
Even though additive semigroups of Q have many interesting and diverse arithmetical
and atomic properties, the semigroup algebras they determine have not been explored
in much detail so far. Examples of semigroup algebras of additive submonoids of Q
have sporadically shown up in the literature; nontrivial examples have been given in
[9, Section 1] and [3, Example 2.1]. In this paper, we study semigroup algebras of
additive submonoids of Q≥0 , which we simply call Puiseux algebras. In particular,
we will construct three families of Puiseux algebras exhibiting different factorization
properties.
The content of this paper is organized as follows. In Section 2 and Section 3, as
an excuse to settle down notation, we introduce most of the main general concepts we
will be using later. In Section 4, we show that primary Puiseux monoids determine
atomic Puiseux algebras. Then, in the first part of Section 5, we find an infinite family
of antimatter Puiseux algebras over algebraically closed fields. Finally, we construct
another infinite family of antimatter Puiseux algebras over any given field of finite
characteristic.
2. Notation and Background
Throughout this sequel, we set N := {1, 2, . . . } and N0 := N ∪ {0}. We also let
P denote the set of prime numbers. In addition, for X ⊆ R and r ∈ R, we define
X≥r := {x ∈ X | x ≥ r}; with a similar spirit we use the notations X<r and X>r .
If q ∈ Q>0 , then we call the unique a, b ∈ N such that q = a/b and gcd(a, b) = 1
the numerator and denominator of q and denote them by n(q) and d(q), respectively.
Finally, for S ⊆ Q>0 , we set
n(S) := {n(s) | s ∈ S}

and

d(S) := {d(s) | s ∈ S}.

Withing the scope of our exposition, each monoid is tacitly assumed to be commutative and cancellative. Unless we specify otherwise, monoids are written additively.
For a monoid M, we let U(M) denote the set of units (i.e., invertible elements) of M.
When U(M) = {0}, we say that M is reduced. Also, we set M • := M \ {0}. The
quotient group of M is denoted by q(M), i.e.,
q(M) := {x − y | x, y ∈ M}.
For S ⊆ M, we write hSi for the smallest (under inclusion) submonoid of M containing
S. If M = hSi, then we call S a generating set of M. In addition, M is called
finitely generated if it can be generated by a finite set, while M is called cyclic if it can
be generated by a singleton. Further basic definitions and concepts on commutative
cancellative monoids can be found in [18, Chapter 2].

ATOMIC AND ANTIMATTER SEMIGROUP ALGEBRAS

3

If y, z ∈ M, then y divides z in M, in symbols, y |M z, if there exists x ∈ M such
that z = x+ y. An element p ∈ M \ U(M) is called prime provided that for all x, y ∈ M
with p |M x + y, either p |M x or p |M y. In addition, an element a ∈ M \ U(M) is an
atom if for any x, y ∈ M such that a = x + y either x ∈ U(M) or y ∈ U(M). The set
of all atoms of M is denoted by A(M), and M is atomic if M = hA(M)i. On the other
hand, we call M antimatter if M does not contain any atoms, i.e., A(M) is empty. A
subset I of M is an ideal of M if I + M ⊆ I. The ideal I is principal if I = x + M for
some x ∈ M. The monoid M satisfies the ascending chain condition on principal ideals
(or ACCP) if each increasing sequence of principal ideals of M eventually stabilizes; in
this case, M is called an ACCP monoid. It is well known that each ACCP monoid is
atomic [8, Proposition 1.1.4].
A multiplicative monoid F is free abelian on a subset P of F provided that every
element a ∈ F can be written uniquely in the form
Y
a=
pvp (a) ,
p∈P

where vp (a) ∈ N0 and vp (a) > 0 only for finitely many p ∈ P . For each set P , there
exists a unique (up to canonical isomorphisms) monoid F that is free abelian on P .
The fundamental theorem of arithmetic states that the multiplicative monoid N is
free abelian on P. In this case, we can extend vp to Q≥0 for each p ∈ P by setting
vp (r) := vp (n(r)) − vp (d(r)) for each r ∈ Q>0 and vp (0) = ∞. The map vp is called
the p-adic valuation on Q≥0 . The next two properties of the p-adic valuation, used
throughout the paper without explicit mention, are easy to verify:
vp (rs) = vp (r) + vp (s) for all r, s ∈ Q≥0 ;
vp (r + s) ≥ min{vp (r), vp(s)} for all r, s ∈ Q≥0 .
Suppose now that M is a reduced monoid. The free abelian monoid on A(M),
denoted by Z(M), is called the factorization monoid of M, and elements in Z(M) are
called factorizations. If z = a1 . . . an ∈ Z(M) for some n ∈ N0 and a1 , . . . , an ∈ A(M),
then n is the length of z and is denoted by |z|. The factorization homomorphism of M
is the unique homomorphism φ : Z(M) → M satisfying φ(a) = a for all a ∈ A(M). For
each x ∈ M, the set Z(x) := φ−1 (x) ⊆ Z(M) is the set of factorizations of x. Clearly,
the monoid M is atomic if and only if Z(x) is nonempty for all x ∈ M. Finally, M is
called an FF -monoid if Z(x) is finite for all x ∈ M.
Proposition 2.1. [8, Proposition 2.7.8(4)] Finitely generated monoids are FF-monoids.
3. Semigroup Algebras of Puiseux Monoids
A Puiseux monoid is an additive submonoid of Q≥0 . Every additive submonoid of
Q which is not a group is isomorphic to a Puiseux monoid [11, Theorem 2.9]. The
following result will be referenced later.

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F. GOTTI

Proposition 3.1. [15, Theorem 3.10] A Puiseux monoid is atomic provided that it does
not have 0 as a limit point.
For a stronger version of Proposition 3.1, see [14, Proposition 4.5]. The atomic
structure of Puiseux monoids has been studied in [15], [16], and [17], where many
families of atomic and non-atomic Puiseux monoids have been introduced.
A Puiseux monoid M is primary if there exist an infinite subset P of P and a subset S
of Q>0 with pairwise distinct denominators such that M = hSi and d(S) = P . Primary
Puiseux monoids are atomic. Indeed, we have the following stronger result.
Theorem 3.2. [16, Theorem 5.5] Submonoid of a primary Puiseux monoid are atomic.
For a ring R and a semigroup S, consider the set R[X; S] comprising all functions
f : S → R satisfying that {s ∈ S | f (s) 6= 0} is finite. We shall conveniently represent
an element f ∈ R[X; S] by
f=

X
s∈S

s

f (s)X =

n
X

f (si )X si ,

i=1

where s1 , . . . , sn are precisely those s ∈ S satisfying that f (s) 6= 0. Addition and multiplication on R[X; S] are defined as for polynomials. Under these operations, R[X; S] is
a ring, which is called the semigroup ring of S over R. Following Gilmer [11], we will
write R[S] instead of R[X; S]. When R is a field, R[S] naturally becomes an algebra,
which is called the semigroup algebra of S over R. As in this paper we are mainly
concerned with semigroup algebras over Puiseux monoids, the next definition will save
us from repetitive long descriptions of our main objects.
Definition 3.3. If F is a field and M is a Puiseux monoid, then we say that F [M] is
a Puiseux algebra.
Let F [M] be a Puiseux algebra. We say that f = α1 X q1 + · · · + αk X qk ∈ F [M] \ {0}
is represented in canonical form if αi 6= 0 for i = 1, . . . , k and q1 > · · · > qk . Observe
that any element of F [M] has a unique representation in canonical form. In this case,
deg(f ) := q1 is called the degree of f . Clearly, one has that
deg(f g) = deg(f ) + deg(g)
for all nonzero elements f and g of F [M]. As it is customary for polynomials, we say
that f is a monomial if k = 1,.
Since M is commutative, cancellative, and torsion free, it follows that F [M] is an
integral domain [11, Theorem 8.1]. Also, the fact that M is cancellative and torsionfree forces the set of units of F [M] to be F × [11, Theorem 11.1]. Additionally, F [M] is
Noetherian if and only if M is finitely generated [11, Theorem 7.7].

ATOMIC AND ANTIMATTER SEMIGROUP ALGEBRAS

5

Proposition 3.4. [11, Theorem 7.2(2)] Let φ : M → M ′ be a monoid homomorphism,
and let φ∗ : F [M] → F [M ′ ] be the algebra homomorphism given by X s 7→ X φ(s) . If φ is
injective (resp., surjective), then φ∗ is injective (resp., surjective).
It is not hard to see that, unless M ∼
= (N0 , +), no monomial of F [M] can be a prime
element; this is a consequence of the trivial fact that non-cyclic Puiseux monoids do not
contain prime elements. Furthermore, a monomial in F [M] is irreducible if and only if
its degree is an atom of M [7, Lemma 3.1]. Thus, the next result holds.
Proposition 3.5. If F [M] contains no irreducible elements, then M is antimatter.
4. Atomic Puiseux Algebras
Our purpose in this section is to construct an infinite family of non-isomorphic
Puiseux algebras whose elements can be written as products of irreducibles.
Definition 4.1. An integral domain R is atomic if each nonzero non-unit element of
R can be expressed as a product of irreducibles.
In addition, recall that an integral domain R satisfies the ascending chain condition
on principal ideals (ACCP) if every increasing chain of principal ideals of R eventually
stabilizes.
Given a Puiseux algebra F [M], it is not hard to verify that the atomicity of M
is a necessary condition for F [M] to be atomic. Conversely, it is natural to ask for
which atomic Puiseux monoids M the algebra F [M] is also atomic. It was proved
in [3, Section 2] that, for any field F and M = h1/p | p ∈ Pi, the algebra F [M]
satisfies the ACCP. Motivated by this example, we will prove that, for each primary
Puiseux monoid M, the algebra F [M] (over any field F ) satisfies the ACCP and, by
Proposition 4.2, is atomic. The following result is standard and a direct consequence
of [3, Theorem 1.3(a)].
Proposition 4.2. If R is an integral domain satisfying the ACCP, then R is atomic.
Given a monoid M and x ∈ M, we let AM (x) denote the set of atoms dividing x in
M. The next lemma will be used in the proof of Proposition 4.4
Lemma 4.3. Let M be a Puiseux monoid, and let x ∈ M. Then Z(x) is finite if and
only if AM (x) is finite.
Proof. To prove the forward implication, take x ∈ M such that |Z(x)| < ∞. Then
observe that each atom in AM (x) must appear in at least one factorization in Z(x).
Thus, |Z(x)| < ∞ implies that |AM (x)| < ∞. Conversely, assume that x ∈ M satisfies
that |AM (x)| < ∞. Then AM (x) = {a1 , . . . , an } for some a1 , . . . , an ∈ A(M). Consider
the submonoid M ′ = ha1 , . . . , an i of M. As A(M) ∩ M ′ ⊆ A(M ′ ), it follows that
A(M ′ ) = {a1 , . . . , an }. By Proposition 2.1, M ′ is an FF-monoid. Now the fact that

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F. GOTTI

AM (x) = AM ′ (x) implies that the sets of factorizations of x in M and M ′ coincide.
Hence |Z(x)| < ∞, which completes the proof.

Proposition 4.4. Each primary Puiseux monoid satisfies the ACCP.
Proof. Let {pn } be a strictly increasing sequence of prime numbers, and let {an } be a
sequence of positive integers with pn ∤ an . Now set M = han /pn | n ∈ Ni. It is not hard
to verify that for all x ∈ M there exist k, n ∈ N0 and 0 ≤ αi < pi such that
x=n+

(4.1)

k
X

αi

i=1

ai
.
pi

Let us proceed to check that the sum decomposition in (4.1) is unique. To do this,
suppose that k ′ , m ∈ N0 and 0 ≤ βi < pi satisfy that
(4.2)

n+

k
X
i=1

k′

X ai
ai
αi = m +
βi .
pi
pi
i=1

Suppose, without loss of generality, that k = k ′ . After isolating (αi − βi )ai /pi in (4.2)
and applying the pi -adic valuation, we obtain that pi | αi − βi , which implies that
αi = βi for each i ∈ {1, . . . , k}. As a consequence, n = m. Since each element of M
•
can be written in a unique way as in (4.1), for each x ∈ M<1
, one has |Z(x)| = 1 and,
by Lemma 4.3, |A(x)| < ∞.
Let us assume, by way of contradiction, that M does not satisfy the ACCP. Then
there exists a strictly decreasing sequence {qn } of elements in M such that
(4.3)

q1 + M ( q2 + M ( · · · .

Now suppose that in the unique sum decomposition of q1 as in (4.1), N is the integer and
pn1 , . . . , pnk are the distinct prime denominators of the atoms with nonzero coefficients.
By (4.3), for each n ∈ N there exists rn+1 ∈ M such that qn = qn+1 +rn+1 and, therefore,
n
X
i=1

ri+1 ≤ qn+1 +

n
X

ri+1 = q1

i=1

for each n ∈ N. Thus, lim rn = 0. In this case, there exist t ∈ N and (possibly
repeated) prime numbers pm1 , . . . , pmt satisfying that |{pm1 , . . . , pmt }| > N + k and
ami /pmi ∈ A(ri+1) for each i = 1, . . . , t. P
Take zi ∈ Z(ri ) containing thePatom ami /pmi ,
and take zt+1 ∈ Z(qt+1 ). As q1 = qt+1 + ti=1 ri+1 , we have z = zt+1 + ti=1 zi ∈ Z(q1 ).
By the uniqueness of the sum decomposition in (4.1), the factorization z contains at
least d(a) copies of each atom a for which
d(a) ∈ P := {pm1 , . . . , pmt } \ {pn1 , . . . , pnk }.
P
Since |P | > N, it follows that N ≥ a∈P d(a)a ≥ |P | > N, which is a contradiction.
Hence M satisfies the ACCP. The fact that M was taken to be an arbitrary Puiseux
monoid completes the proof.


ATOMIC AND ANTIMATTER SEMIGROUP ALGEBRAS

7

The next standard result is used to prove Theorem 4.6. We include its proof here for
the sake of completeness.
Proposition 4.5. If a reduced monoid M satisfies the ACCP, then the semigroup
algebra F [M] also satisfies the ACCP.
Proof. Let (f1 ) ⊆ (f2 ) ⊆ · · · be an ascending chain of principal ideals of F [M]. Then
the sequence {deg(fn ) + M} of subsets of M is an ascending chain of principal ideals,
which stabilizes because M satisfies the ACCP. As M is reduced, there exists N ∈ N
such that deg(fn ) = deg(fN ) for all n ≥ N. This, along with the fact that fn divides
fN in F [M] for each n ≥ N implies that (fn ) = (fN ) for all n ∈ N.

We are now in a good position to find a family of atomic Puiseux algebras.
Theorem 4.6. For each field F , there is an infinite family of pairwise non-isomorphic
atomic Puiseux algebras over F .
Proof. Let {Pn } be a sequence of pairwise disjoint infinite subsets of P. For each n ∈ N,
consider the primary Puiseux monoid
Mn := h1/p | p ∈ Pn i.
We claim that the members of the family F := {F [Mn ] | n ∈ N} of Puiseux algebras
are atomic and pairwise non-isomorphic. The atomicity follows by combining Proposition 4.4, Proposition 4.5, and Proposition 4.2.
To show that the algebras in F are pairwise non-isomorphic, suppose, by way of
contradiction, that F [Mj ] ∼
= F [Mk ] for distinct j, k ∈ N. Let φ : F [Mj ] → F [Mk ] be
an algebra isomorphism. Since φ fixes F , it follows that φ(X q ) ∈
/ F for any q ∈ Mj• .
This implies that deg(φ(X)) ∈ Mk• . As Pj is unbounded there exists p ∈ Pj such that
p > n(deg(φ(X))). Observe that

1 p 
1 
deg φ(X) = deg φ X p
(4.4)
= p deg φ X p .
Because p ∈
/ Pk , it does not divide any element in d(Mk• ). Thus, it follows from (4.4)
that p | n(deg φ(X)), which contradicts that p > n(deg(φ(X))). Hence the Puiseux
algebras in F are atomic and pairwise non-isomorphic, as desired.

5. Antimatter Puiseux Algebras
Let M be a monoid, and recall that q(M) denotes the quotient group of M. An
element x ∈ q(M) is called integral over M if nx ∈ M for some n ∈ N. The subset M
of q(M) consisting of all integral elements over M is called the integral closure of M.
If M = M, then M is called integrally closed. Here we exhibit two characterizations of
integrally closed Puiseux monoids.

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F. GOTTI

Theorem 5.1. [11, Theorem 13.5] Let M be a Puiseux monoid containing 1. The
following conditions are equivalent:
(1) M is integrally closed;
(2) q(M) ∩ Q≥0 = M;
(3) M is the union of an ascending sequence of cyclic additive submonoids of Q.
Example 5.2. Since the Puiseux monoid Q≥0 satisfies condition (2) of Theorem 5.1,
it is integrally closed. To find an ascending sequence of cyclic submonoids of Q whose
union is Q≥0 , take {pn } to be an enumeration of P and notice that


[
1
Q≥0 =
Pn , where Pn :=
.
n
n
p
.
.
.
p
1
n
n∈N
Recall that an integral domain is called a Bezout domain if the addition of two
principal ideals is again a principal ideal and is called a Prüfer domain provided that
each nonzero finitely generated ideal is invertible. Integrally closed Puiseux monoids
are important because the Puiseux algebras they determine are both Bezout domains
and Prüfer domains.
Theorem 5.3. [11, Theorem 13.5] Let F be a field, and let M be a Puiseux monoid
containing 1. The following conditions are equivalent:
(1) M is integrally closed;
(2) F [M] is a Bezout domain;
(3) F [M] is a Prüfer domain.
Integrally closed Puiseux monoids exhibit extreme atomic structure, i.e., they are
either atomic or antimatter, as the next proposition illustrates.
Proposition 5.4. For an integrally closed nontrivial Puiseux monoid M, the following
two conditions hold.
(1) M is atomic if and only if M is cyclic.
(2) M is antimatter if and only if 0 is a limit point of M.
Proof. Let us argue condition (1). Suppose first that M is atomic. Since M is integrally
closed, Theorem 5.1 ensures the existence of a sequence {qn } of rational numbers such
that hq1 i ⊆ hq2 i ⊆ · · · and M = ∪n∈N hqn i. As M is nontrivial, it must contain an atom
a. Clearly, a = qj for some j ∈ N. Because a is an atom, the fact that a ∈ hqn i for
each n ≥ j forces qn = a for each n ≥ j. Hence M = hai. The backward implication of
condition (1) follows trivially.
Now we verify condition (2). Suppose that M is antimatter. Because M is nontrivial,
it is not a group. Thus, it cannot be atomic and antimatter simultaneously. Then M
is not atomic. By Proposition 3.1, one has that 0 is a limit point of M, and the
direct implication of (2) follows. Conversely, suppose that 0 is a limit point of M,

ATOMIC AND ANTIMATTER SEMIGROUP ALGEBRAS

9

and let the sequence of rationals {qn } be defined as in the previous paragraph. Since
A(M) ⊆ {qn | n ∈ N}, it suffices to check that qn ∈
/ A(M) for any n ∈ N. To see this,
observe that because lim qn = 0, for each n ∈ N there exists m > n such that qm < qn
and so qm strictly divides qn in M. Hence M is antimatter.

Proposition 5.4 specifies, in particular, which integrally closed Puiseux monoids are
antimatter. The following example, on the other hand, exhibits an antimatter Puiseux
monoid that fails to be integrally closed.
Example 5.5. The Puiseux monoid M =: 1/2n , 1/3n | n ∈ N is clearly antimatter.
Suppose, by way of contradiction, that M is integrally closed. By Theorem 5.1, there
exists an ascending sequence {hrn i} of cyclic submonoids of M such that
[
M=
hrn i.
n∈N

Take k ∈ N such that 1/2 ∈ hrk i. It is easy to see that n(rk ) = 1, which implies that
n(rn ) = 1 for each n ≥ k. Let d(rk ) = 2a 3b for some a, b ∈ N0 . Since r = 1/2a+1 +1/3b+1
satisfies that d(r) = 2a+1 3b+1 , it follows that r ∈ M \ hrk i. Now take ℓ ∈ N such that
r ∈ hrℓ i. Then ℓ > k and, therefore, n(rℓ ) = 1. Because r is an integer multiple of rℓ ,
we have that d(rℓ ) = 2c 3d , where c ≥ a + 1 and d ≥ b + 1. Hence
1
1
= 2c−13d−1 c d ∈ hrℓ i ⊆ M.
6
23
So there exist s, t ∈ N and nonnegative coefficients α1 , . . . αt and β1 , . . . , βs with either
αt 6= 0 or βs 6= 0 satisfying that
t

(5.1)

s

1 X αi X βi
=
+
.
6
2i
3i
i=1
i=1

After simplifying if necessary, we can assume that 2 ∤ αi and 3 ∤ βj for i = 1, . . . , t and
j = 1, . . . , s. If t 6= 0, then multiplying (5.1) by 2t one obtains that t = 1. Similarly, if
s 6= 0, then s = 1. The fact that s, t ∈ {0, 1} forces 1/6 ∈ {α1 /2, β1 /3, α1/2 + β1 /3},
which is a contradiction.
Remark 5.6. The fact that 1/6 ∈ M in Example 5.5 could have also been achieved by
verifying first that


n
q(M) =
n ∈ Z and r, s ∈ N0 ,
2r 3s
and then using the integrally closedness of M, along with the fact that 5/6 ∈ M, to
conclude that 1/6 ∈ M.
We are ready now to construct a family of semigroup algebras without irreducible
elements.

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F. GOTTI

Definition 5.7. An integral domain R is called antimatter provided that R has no
irreducible elements.
Antimatter domains have been studied by Coykendall, Dobbs, and Mullins in [6]. As
the next proposition reveals, integrally closed Puiseux monoids can be used to construct
antimatter Puiseux algebras over algebraically closed fields.
Proposition 5.8. Let F be an algebraically closed field, and let M be a nontrivial
integrally closed Puiseux monoid. Then the following conditions hold.
(1) If M is not antimatter, then F [M] ∼
= F [X].
(2) If M is antimatter, then F [M] is an antimatter Bezout domain.
Proof. If M is not antimatter, then Proposition 5.4 ensures that M is isomorphic to
(N0 , +). Then Proposition 3.4 guarantees that F [M] ∼
= F [X], and condition (1) follows.
On the other hand, condition (2) is a restatement of [2, Theorem 1].

Example 5.9. The integrally closedness of M in Proposition 5.8 is necessary for condition (1) to hold. To see this, consider the Puiseux monoid M = h1/3, 1/2n | n ∈ Ni.
2
3
It is not antimatter as 1/3 is an atom. However, (X 1/2 ) ( (X 1/2 ) ( (X 1/2 ) ( · · · is
an ascending chain of ideals of F [M] that does not stabilize, while F [X] happens to be
a Noetherian domain.
On the other hand, we wonder whether the integrally closedness of M is necessary
for condition (2) in Proposition 5.8 to hold.
Question 5.10. Is there a non-integrally closed and antimatter Puiseux monoid such
that the algebra F [M] is antimatter over any (or some) algebraically closed field F ?
Now we construct an infinite family of antimatter Puiseux algebras.
Theorem 5.11. Let F be an algebraically closed field. There are infinitely many nonisomorphic antimatter Puiseux algebras over F .
Proof. Consider the set of primes P := P \ {char(F )} and, for each p ∈ P , define
Mp = h1/pn | n ∈ Ni. Now consider the family of Puiseux algebras
F := {F [Mp ] | p ∈ P }.
Clearly, each Mp is an antimatter monoid. In addition, because 1/pn+1 divides 1/pn in
Mp , it follows that Mp is the union of an ascending sequence of cyclic submonoids of Q,
namely, Mp = ∪n∈N h1/pn i. Thus, each Mp is integrally closed by Theorem 5.1. Since
F is algebraically closed, Proposition 5.8(2) guarantees that each algebra F [Mp ] is an
antimatter Bezout domain.
Finally, let us verify that the algebras in F are pairwise non-isomorphic. Suppose, by
way of contradiction, that there exists p, q ∈ P with p 6= q such that F [Mp ] ∼
= F [Mq ].
Let φ : F [Mp ] → F [Mq ] be an algebra isomorphism. As in the proof of Theorem 4.6, we

ATOMIC AND ANTIMATTER SEMIGROUP ALGEBRAS

11

j

have that deg(φ(X 1/p )) 6= 0 for every j ∈ N. Take n ∈ N such that n > vp (deg(φ(X))).
Since p 6= char(F ),

1 
1 
vp deg φ(X) = vp pn deg φ X pn
(5.2)
= n + vp deg φ X pn
.

n
Since d(Mq ) consists only of powers of q and deg φ X 1/p
∈ Mq• , it follows that


n
vp deg φ X 1/p
≥ 0. Therefore vp deg φ(X) ≥ n as a consequence of (5.2),
which rises a contradiction. Hence F is a countable collection of pairwise non-isomorphic
antimatter Puiseux algebras, which completes the proof.

Finally, we construct a family of antimatter Puiseux algebra over perfect fields of
finite characteristic, now using certain divisible Puiseux monoids. Let M be a Puiseux
monoid. For n ∈ N and x ∈ M, we say that x is n-divisible if x/n ∈ M, and we say that
M is n-divisible if x is n-divisible for each x ∈ M. Clearly, every n-divisible Puiseux
monoid is antimatter provided that n ≥ 2.
Theorem 5.12. Given a perfect field F of finite characteristic, there are infinitely
many non-isomorphic antimatter Puiseux algebras over F .
Proof. Let p be the characteristic of F . First, we introduce the following class of Puiseux
monoids parameterized by the prime numbers in P \ {p}:


1
Mq :=
i, j ∈ N .
pi q j
Clearly, Mq is antimatter and p-divisible for each q ∈ P \ {p}. Now consider the family
of Puiseux algebra F over the field F determined by the Puiseux monoids Mq , namely,

F = F [Mq ] | q ∈ P \ {p} .

Once again, we can use an argument similar to that one given in the proof of Theorem 4.6
to show that the algebras in F are pairwise non-isomorphic: let φ : Mq → Mq′ be an
algebra isomorphism for distinct q, q ′ ∈ P \ {p}, then take n > vq (deg(φ(X 1/p ))), and
finally notice that
1 
1 
1 
n > vq deg φ(X p
= vq q n deg φ X pqn
= n + vq deg φ X pqn
≥ n,

which is a contradiction. As a result, the Puiseux algebras in the collection F are
pairwise non-isomorphic, as desired.
It only remains to verify that each algebra in F is an antimatter domain. To do so,
take M ∈ F and f (X) = α1 X q1 + · · · + αn X qn ∈ F [M] \ F . As F is a perfect field
of characteristic p, the Frobenious homomorphism x 7→ xp is surjective and, therefore,
for each i = 1, . . . , n, there exists βi ∈ F with αi = βip for some βi ∈ F . Also, the
p-divisibility of M ensures that qi /p ∈ M for 1 ≤ i ≤ n. As a consequence, we have
p
that f (X) = β1 X q1 /p + · · · + βn X qn /p is not irreducible in F [M]. Hence F [M] is
antimatter, which completes the proof.


12

F. GOTTI

Remark 5.13. Theorem 5.12 is a version of [2, Theorem 5(2)], which states that if
D is an antimatter GCD-domain whose quotient field is perfect of finite characteristic,
then D[Q≥0 ] is also an antimatter GCD-domain.
Acknowledgements
During the preparation of this paper, the author was supported by the NSF-AGEP
fellowship.
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Department of Mathematics, UC Berkeley, Berkeley, CA 94720
E-mail address: felixgotti@berkeley.edu

