Weight Ideals Associated to Regular and Log-Linear Arrays

0.2 MB·English

by  J. W. Johnson

Weight Ideals Associated to Regular and
Log-Linear Arrays
Jeremiah W. Johnson
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Dept. of Mathematics and Statistics, Penn State Harrisburg
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2 Middletown, PA 17057

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a
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1 Abstract
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Certain weight-based orders on the free associative algebra R = khx ,...,x i can
] 1 t
A be specified by t×∞ arrays whose entries come from the subring of nonnegative
R elements in a totally ordered field. Such an array A satisfying certain additional
. conditionsproducesapartialorderonRwhichisanadmissibleorderonthequotient
h
t R/I , where I is a homogeneous binomial ideal called the weight ideal associated
a A A
m to the array and whose structure is determined entirely by A. This article discusses
the structure of the weight ideals associated to two distinct sets of arrays whose
[

elements define admissible orders on the associated quotient algebra.
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v
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Key words: Noncommutative Gro¨bner Bases, Gro¨bner Bases, Admissible Orders
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6
.
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1 1 Introduction
:
v
i
X
r Work over the past two decades has extended the theory of Gro¨bner bases
a
to various noncommutative algebras (Green, 2000; Madlener & Reinert, 1997;
Nordbeck,2001;Mora,1994).BeforeaGr¨obnerbasisforanidealofak-algebra
Acanbeconstructed, wherek isafield,anadmissibleorderonamultiplicative
basis of A is required. Following Green (1996), we say that A has a Gr¨obner
basis theory when an admissible order exists on a multiplicative basis of A. In
Hinson (2010), E. Hinson adapted the theory of position-dependent weighted
orders to define a length-dominant partial order on the set of words in the free
associative algebra R = khx ,...,x i, including the trivial word, which pro-
1 t
duces an admissible order on a quotient of R. In this construction, the partial
order on R is specified by a t×∞ array Awhose entries come from the subring
consisting of the positive elements of a totally ordered field, and the quotient
Email address: [email protected] (Jeremiah W. Johnson).
Preprint submitted to Elsevier 1 February 2011
is by a homogenous binomial ideal I whose elements are determined by the
A
partial order given by A. This gives rise to two immediate questions. First,
given an array A that defines an admissible order on a quotient R/I , what
A
is the algebra that is determined, or more specifically, what is the structure
of the ideal I ? Second, given two arrays A and B which define orders ≻
A A
and ≻ on R/I and R/I respectively, even when R/I = R/I it is not
B A B A B
necessarily the case that ≻ =≻ . Under what circumstances does ≻ =≻ ?
A B A B
This paper describes results concerning the first of these two questions for two
distinct families of admissible arrays. In this introductory section, we review
the relevant definitions and results from Hinson (2010) and we make the pre-
ceding general statements precise. Our primary objects of interest are defined
in Definitions 5 and 6. The results on which the remainder of the paper relies
are given in Theorems 7 and 8. In what follows, let R = khx ,...,x i denote
1 t
the free associative algebra, let S denote the positive elements of a totally
>0
ordered field, and let M (S ) denote the set of t×∞ arrays with entries
t×∞ >0
in S . The following two definitions are adopted from Green (1996).
>0
Definition 1 Let B be a k–basis of an algebra A. B is a multiplicative basis
for A if
′ ′ ′
b,b ∈ B ⇒ b·b ∈ B or b·b = 0.
We will have occasion to refer to the nontrivial elements of B, which we denote
by B×.
Definition 2 A total order≻ on a multiplicative basis B of A is an admissible
order on B if
• ≻ is a well-order on B,
• for all b , b , b ∈ B such that b b 6= 0 and b b 6= 0, if b ≻ b , then
1 2 3 1 3 2 3 1 2
b b ≻ b b ,
1 3 2 3
• for all b , b , b ∈ B such that b b 6= 0 and b b 6= 0, if b ≻ b , then
1 2 3 3 1 3 2 1 2
b b ≻ b b , and
3 1 3 2
• for all b , b , b , b ∈ B, if b = b b b , then b (cid:23) b .
1 2 3 4 1 2 3 4 1 3
Commonly used admissible orders for Gr¨obner basis calculations on noncom-
mutative algebras are the left length-lexicographic order or the right length-
lexicographic order (Green, 1996). We specify a position-dependent weighted
order on words in the free algebra using a t × ∞ array to define a weight
function as described in the following definition.
Definition 3 Let A = (a ) ∈ M (S ). A gives a monomial weighting
i,j t×∞ >0
σ : B× → S by
A >0
l−1
σ (x x ···x ) = a
A u0 u1 ul−1 Y uj,j
j=0
2
for a given monomial x x ···x ∈ R. The function σ is the weight
u0 u1 ul−1 A
function associated to A.
Note that for computational convenience we index the columns of an array
starting with 0 rather than 1. When the array A is clear, we will suppress it
from the notation and write the associated weight function σ simply as σ.
A
For the remainder of this section, fix an array A ∈ M (S ) and associated
t×∞ >0
weight function σ. In order to discuss the weight of the product of two words,
we identify a translated version of the weight function associated to A by
l−1
σ (x x ···x ) = a ,
k u0 u1 ul−1 Y uj,j+k
j=0
where k ∈ N. We consider σ(ω) = σ (ω) = σ (ω). Let |ω| denote the length
A A,0
of ω. Given ω and λ such that |ω| = k,
σ(ωλ) = σ(ω)·σ (λ).
k
This gives rise to the following equivalence relation.
Definition 4 Define the relation ≻ on B× by
σ
ω ≻ ω ⇐⇒ |ω | > |ω |, or |ω | = |ω | and σ(ω ) > σ(ω ).
1 σ 2 1 2 1 2 1 2
Let Γ denote the set of pure homogeneous binomial differences ω −ω , where
1 2
ω , ω ∈ B×, |ω | = |ω |, and σ(ω ) = σ(ω ).
1 2 1 2 1 2
Definition 5 The ideal I = hΓi is the weight ideal associated to A.
A
Definition 6 A is an admissible array if for every pair ω , ω ∈ B× with
1 2
|ω | = |ω |,
1 2
(1) for all k ≥ 0, if σ (ω ) > σ (ω ), then σ (ω ) > σ (ω ), and
k 1 k 2 k+1 1 k+1 2
(2) for all k ≥ 0, if σ (ω ) = σ (ω ), then σ (ω ) = σ (ω ).
k 1 k 2 k+1 1 k+1 2
The following theorem illustrates that the second part of Definition 6 is in fact
unnecessary.
Theorem 7 Let A ∈ M (S ) be an array with associated weight function
t×∞ >0
σ. The following are equivalent:
(1) A is an admissible array;
(2) for all k ≥ 0 and for all ω , ω ∈ B× such that |ω | = |ω |, σ (ω ) > σ (ω )
1 2 1 2 k 1 k 2
if and only if σ (ω ) > σ (ω ).
k+1 1 k+1 2
3
Admissible arrays define an admissible order on the quotient R/I .
A
Theorem 8 An array A ∈ M (S ) with associated weight function σ is
t×∞ >0
an admissible array if and only if ≻ is an admissible order on B ⊆ R/I ,
σ σ A
where B is the image of B in R/I under the projection R → R/I .
σ A A
Definition 9 A is said to be degenerate if there exists i, j, 1 ≤ i 6= j ≤ t,
such that σ(x ) = σ(x ).
i j
We will assume in what follows that all arrays considered are nondegenerate,
for if σ(x ) = σ(x ) for some i,j ∈ {1,...,t} where i 6= j, then x −x ∈ I
i j i j A
and khx ,...,x i/I ≃ khx ,...,x ,x ,...,x i/hI′ i where A′ is the array
1 t A 1 i−1 i+1 t A
obtained from A by deleting the ith row.
2 Weight Ideals Associated to Regular Arrays
In Hinson (2010), E. Hinson described two sets of admissible arrays. We begin
by studying the first of these, the set of regular arrays.
Definition 10 An array A is regular if A has rank 1.
The set of linear arrays is a subset of the set of regular arrays which will be
used later on to construct the set of log-linear arrays.
Definition 11 An array A is linear if for all i ≥ 1, A = d·A for some
(i) (i−1)
fixed d ∈ S . The fixed scalar d is referred to as the slope of the array.
>0
Example 12 The array
2 6 18 ···
A = 3 9 27 ···
 
 
 
4 12 36 ···
 
is a linear array with slope d = 3.
The weight ideal associated to a regular array contains the commutator ideal
C = hx x −x x |1 ≤ i 6= j ≤ ti, and thus is never trivial (Hinson, 2010).
i j j i
Definition 13 The support of a word ω is the set
supp(ω) = {x |i ∈ {1,...,t} and x occurs in ω}.
i i
Definition 14 The frequency of x in ω is the number of times that x occurs
in ω and is written #(x,ω).
4
Definition 15 Let f ∈ R and G = {g ,g ,...} ⊂ R. We say that f is an
1 2
algebraic consequence of G if f = c u g v , where c ∈ k, u , v ∈ R, and
g∈G i i i i i i i
P
only finitely many c 6= 0.
i
Suppose A is a regular array with first column [a ,...,a ]T, where a ∈
1,0 t,0 i,0
N and at least one of (a ,a ) 6= 1, where (a ,a ) denotes the greatest
i,0 j,0 i,0 j,0
common divisor of a and a and 1 ≤ i 6= j ≤ t. Let ω = x ···x and
i,0 j,0 1 u0 ul−1
ω = x ...x ∈ B such that ω −ω ∈ I . Then we have
2 v0 vl−1 1 2 A
l−1 l−1
a = a ,
Y uii Y vii
i=0 i=0
and each a and a can be written as scalar multiples of a and a
uii vii ui0 vi0
respectively:
l−1 l−1
d a = d a .
Y i ui0 Y i vi0
i=0 i=0
Factoring out and canceling the common d ’s reduces the equation to
i
l−1 l−1
a = a . (1)
Y ui0 Y vi0
i=0 i=0
Equation (1) does not depend on the how the variables were ordered in ω
1
and ω ; in particular, by factoring out and canceling any terms a = a
2 ui0 vj0
common to both sides of the equation, one obtains the reduced expression
n n
a = a . (2)
Y uk0 Y vk0
k=0 k=0
In this expression, aum0 6= avm′0 for all um and vm′. Note that we have not
cancelled any common divisors of the a , we have only cancelled those a ’s
ui,0 ui,0
and a ’s for which a = a . Since each a and a corresponds to the
vj,0 ui,0 vj,0 um0 vm′0
weight assigned to an individual letter in {x ,...,x }, this equation describes
1 t
a homogeneous binomial difference ω′ −ω′ ∈ I in which no letter that occurs
1 2 A
in ω′ will occur in ω′.
1 2
Definition 16 A homogeneous binomial difference ω −ω ∈ I for which
1 2 A
supp(ω ) supp(ω ) = ∅
1 \ 2
will be referred to as a homogeneous binomial difference of disjoint support.
Homogeneous binomial differences of disjoint support may arise as algebraic
consequences of other homogeneous binomial differences of disjoint support.
5
For example, suppose
2 4 8 ···
 
3 6 12 ···
 
A =  .
 
4 8 16 ···
 
 
6 12 24 ···
 
This array A is linear with slope 2. Consider the homogeneous binomial dif-
ference x x x x −x x x x . Since
3 2 3 2 4 1 4 1
σ(x x x x ) = σ(x x x x ) = 9216,
3 2 3 2 4 1 4 1
we must have x x x x − x x x x ∈ I . Neither word in this homogeneous
3 2 3 2 4 1 4 1 A
difference shares a letter with the other, so x x x x −x x x x is a homoge-
3 2 3 2 4 1 4 1
neous binomial difference of disjoint support. Furthermore,
x x x x −x x x x = (x x −x x )x x +x x (x x −x x ),
3 2 3 2 4 1 4 1 3 2 4 1 3 2 4 1 3 2 4 1
so x x x x −x x x x is a homogeneous binomial difference of disjoint sup-
3 2 3 2 4 1 4 1
port which arises as an algebraic consequence of a homogeneous binomial
difference of disjoint support consisting of words of lesser length.
Definition 17 Let ω −ω be a homogeneous binomial difference of disjoint
1 2
support. ω −ω is minimal if any expression
1 2
n
ω −ω = α (u −v )β
1 2 X i i i i
i=1
for ω − ω as a sum of homogeneous binomial differences has at least one
1 2
difference u − v such that |u | = |v | = |w |. M will be used to denote
i i i i 1 A
the set of minimal length homogeneous binomial differences of disjoint support
associated to A.
Inother words, a minimal homogeneous binomial difference of disjoint support
is one which cannot be realized as an algebraic consequence of homogeneous
binomial differences of disjoint support consisting of words of lesser length.
Any element of I may be decomposed over the set of commutators {x x −
A i j
x x |1 ≤ i 6= j ≤ t}andthesetofhomogeneousbinomialdifferences ofdisjoint
j i
support.
Lemma 18 Let ω − ω be a homogeneous binomial difference in I . Then
1 2 A
ω − ω = n α (u − v )β , where each homogeneous binomial difference
1 2 Pi=1 i i i i
u − v , 1 ≤ i < n is a commutator and u − v is a homogeneous binomial
i i n n
difference of disjoint support.
6
PROOF. Suppose ω − ω ∈ I . We proceed by induction. The base case
1 2 A
when l = 2 is established trivially. Assume now that the induction hypothesis
holds for homogeneous binomial differences consisting of words of length l−1
and suppose |ω | = |ω | = l. Write ω = x ...x and ω = x ...x .
1 2 1 u0 ul−1 2 v0 vl−1
Let i ∈ {0,...,l −1} be the least value for which x = x (if no such value
u0 vi
exists, we are done). By inserting the expression
−x ...x x x x ...x +x ...x x x x ...x ,
v0 vi−2 vi vi−1 vi+1 vl−1 v0 vi−2 vi vi−1 vi+1 vl−1
we obtain
ω −x ...x x x x ...x +x ...x x x x ...x −ω ,
1 v0 vi−2 vi vi−1 vi+1 vl−1 v0 vi−2 vi vi−1 vi+1 vl−1 2
which is equal to
ω −x ...x x x x ...x +
1 v0 vi−2 vi vi−1 vi+1 vl−1
x ...x x x −x x x ...x . (3)
v0 vi−2 (cid:16) vi vi−1 vi−1 vi(cid:17) vi+1 vl−1
In the second term in expression (3), x occurs in the i − 1st position. The
vi
third and fourth terms in Equation 3 have been expressed as (left and right)
multiples of the commutator x x −x x . Iterating this process i times
vi vi−1 vi−1 vi
results in the expression
i−1
ω −x x ...x x ...x + α (x x −x x )β , (4)
1 vi v0 vi−1 vi+1 vl−1 X k vi vi−k vi−k vi k
k=1
where α = x ...x and β = x ...x .
k v0 vi−k−1 k vi−k+1 vl−1
Since x = x , the difference of the first two terms in 4 can be rewritten as
u0 vi
x x ...x −x ...x .
u0 (cid:16) u1 ul−1 v1 vl−1(cid:17)
The expression in parentheses consists of monomials of length l − 1 which
is an algebraic consequence of the commutators and a homogeneous binomial
differenceofdisjoint support.Rearrangingandrenamingtermsasneededgives
2
the desired result.
Theorem 19 Let A be a regular array. The weight ideal I associated to a
A
regular array A is generated by the union of the set of commutators {x x −
i j
x x |1 ≤ i 6= j ≤ t} and M .
j i A
7
PROOF. Fix a homogeneous binomial difference ω −ω ∈ I . By iterating
1 2 A
thealgorithmdescribed intheproofofLemma 18,ω −ω ∈ I canbereduced
1 2 A
to an algebraic consequence of the commutators plus a single, perhaps trivial,
homogeneous binomial difference of disjoint support ω′ − ω′. To see this,
1 2
note that each iteration of the algorithm produces in the sum a difference of
commutators and a homogeneous binomial difference of shorter length than
in the previous iteration in which a letter common to each word has been
extracted. We may continue the algorithm until either the next iteration is
over a commutator or there are no common letters to extract. In the first
case, we are done, and in the second case, if ω′ −ω′ is minimal, we are also
1 2
done. If ω′ − ω′ is not a minimal homogeneous binomial difference, then by
1 2
definition it is an algebraic consequence of minimal homogeneous binomial
2
differences of disjoint support.
Having obtained a description of the generators of I , we will next show that
A
when A is regular, I is finitely generated. We include the following lemma
A
to describe the means by which a disjoint homogeneous binomial difference
which contains another difference as scattered subwords can be decomposed
over that subdifference.
Lemma 20 Let ω −ω ∈ M and suppose λ −λ is a homogeneous binomial
1 2 A 1 2
difference of disjoint support such that ω occurs as a scattered subword in λ
1 1
and ω occurs as a scattered subword in λ . Then
2 2
n
λ −λ = (ω −ω )α+ω (α−β)+ α (γ −ζ )β ,
1 2 1 2 2 X i i i i
i=1
where α−β is a homogeneous binomial difference of disjoint support and γ −ζ
i i
is a commutator for each i, 1 ≤ i ≤ n.
PROOF. The algorithm of Lemma 18 may be modified to move any letter
thatoccursinawordinahomogeneousbinomialdifferenceinI eitherforward
A
or backwards to the desired position, resulting in a decomposition
n
λ −λ = ω α−ω β + α (γ −ζ )β ,
1 2 1 2 X i i i i
i=1
where γ −ζ is a commutator, 1 ≤ i ≤ n. The result then follows.
i i
Theorem 21 Let A be a regular array. The associated weight ideal I is
A
finitely generated.
PROOF. By Theorem 19, I is generated by the union of the set of commu-
A
tatorsandthesetM ofminimalhomogeneousbinomialdifferencesofdisjoint
A
8
support. The set of commutators is clearly finite. It remains to demonstrate
that M is also finite. Assume the contrary. Then there exists some parti-
A
tion of X = {x ,...,x } into two sets X , X such that there are infinitely
1 t 1 2
many minimal disjoint homogeneous binomial differences ω − ω in which
1 2
supp(ω ) ⊆ X and supp(ω ) ⊆ X . Let D = {ω − ω ∈ M |supp(ω ) ∈
1 1 2 2 1 2 A 1
X , supp(ω ) ∈ X } and let ω −ω ∈ D such that |ω | ≤ |λ | for any λ that
1 2 2 1 2 1 1 1
occurs in a homogeneous binomial difference λ −λ ∈ D. Consider the follow-
1 2
ingthreesets:D(ω ) = {λ −λ ∈ D : ω occurs as a scattered subword in λ },
1 1 2 1 1
D(ω ) = {λ −λ ∈ D : ω occurs as a scattered subword in λ }, and D(0) =
2 1 2 2 2
{λ − λ ∈ D : neither ω nor ω occur as scattered subwords in λ and λ }.
1 2 1 2 1 2
Note that D = {ω −ω }∪D(ω )∪D(ω )∪D(0). Furthermore, these sets are
1 2 1 2
disjoint. If ω were to occur as a scattered subword in λ and ω occurs as a
1 1 2
scattered subword of λ , then Lemma 20 shows that λ − λ is an algebraic
2 1 2
consequence of commutators, ω −ω , and perhaps some other homogeneous
1 2
binomial difference in D consisting of words of length less than |λ |; that is,
1
λ −λ is not minimal. Thus, these sets form a partition of D and so at least
1 2
one of D(ω ), D(ω ), and D(0) must be infinite.
1 2
Now let λ −λ ∈ D(ω ) and suppose |λ | > |ω |. Since λ does not contain ω
1 2 2 1 1 1 1
as a scattered subword, the number of occurrences k of some variable x in λ
i i 1
must beless thanin ω , so the number of occurrences k of some other variable
1 j
x must be greater than the number of occurrences in ω . Suppose D(ω ) is
j 1 2
infinite. Then there exists a difference λ′ − λ′ ∈ D(ω ) with |λ′| > |λ |,
1 2 2 1 1
and furthermore, neither λ nor ω can occur as scattered subwords in λ′.
1 1 1
Thus the number of occurrences ki′ of another variable xi′ must be less than
in ω1, and so the number of occurrences kj′ of another variable xj′ must be
greater than in ω . This indicates that D(ω ) cannot be infinite: for some l,
1 2
any homogeneous binomial difference γ −γ ∈ D(ω ) such that |γ | > l must
1 2 2 1
¯
have a first word which contains as a scattered subword some word λ which
1
¯ ¯
previously occurred in a homogeneous binomial difference λ − λ ∈ D(ω )
1 2 2
and is thus not minimal. The same argument, mutatis mutandis, shows that
D(ω ) is also finite.
1
Consider, then, the set D(0). Let ω′ −ω′ ∈ D(0) be such that |ω′| ≤ |λ | for
1 2 1 1
any λ −λ ∈ D(0). Note that ω′ −ω′ must consist of words at least as long
1 2 1 2
as ω , and furthermore, in both ω′ and ω′ some variables x and x must
1 1 2 k1 k2
occur less often than in ω and ω respectively. We may partition D(0) into
1 2
sets D(ω′), D(ω′), and D(0′) which form a partition of D(0). As above, these
1 2
sets form a partition of D(0), and following the argument above, both D(ω′)
1
and D(ω′) are finite. Consider then D(0′), which must be infinite, and select a
2
difference ω′′−ω′′ ∈ D(0′) such that |ω′′| ≤ |λ | for any λ −λ ∈ D(0′). Again
1 2 1 1 1 2
ω′′−ω′′ must consist of words at least as long as ω′, and furthermore, in both
1 2 1
ω1′′ and ω2′′ some variables xk1′ and xk2′ must occur less often than in ω1′ and ω2′
respectively. Continuing this partitioning process ad infinitum is impossible:
forsomel, anydifference γ −γ such that|γ | > l must containtheoccurrence
1 2 1
9
¯
of some λ , i ∈ {1,2}, which previously occurred in a homogeneous binomial
i
¯ ¯
difference in λ − λ ∈ D(0) as a scattered subword. Thus D(0) cannot be
1 2
infinite, and so M is finite and I must be finitely generated. 2
A A
We have the following corollaries. Corollary 23 gives a description of those ho-
mogeneous binomial differences in the commutator ideal. Note that necessity
in Corollary 23 was proved in Hinson (2010).
Corollary 22 Let A be a regular array with pairwise-coprime first column
entries. Then I = C, where C denotes the commutator ideal.
A
PROOF. Since the entries in the first column of Aare pairwise-coprime, M
A
2
is trivial.
Corollary 23 Let A be a regular array with pairwise-coprime first column
entries, and suppose ω , ω ∈ B with |ω | = |ω | = l. Then ω −ω ∈ I ⇐⇒
1 2 1 2 1 2 A
supp(ω ) = supp(ω ) and #(x ,ω ) = #(x ,ω ) for all x ∈ supp(ω ) =
1 2 i 1 i 2 i 1
supp(ω ).
2
PROOF. To prove sufficiency, let
a d a ··· d a ···
 1,0 1 1,0 n 1,0 
a d a ··· d a ···
 2,0 1 2,0 n 2,0 
A =  . . . .
 . . . 
 . . . 
 
 
a d a ··· d a ···
 t,0 1 t,0 n t,0 
Assume that ω −ω ∈ I . Then
1 2 A
l−1 l−1
σ(ω ) = σ(ω ) ⇒ a = a .
1 2 Y uii Y vii
i=0 i=0
Expressing each weight as a multiple of a first-column entry and canceling the
d ’s common to each side of the equation gives
i
l−1 l−1
a = a . (5)
Y ui0 Y vi0
i=0 i=0
Since thefirst columnentries ofAarepairwise-coprime, equality canonlyhold
in Equation (5) when there exists a bijection between {a }l−1 and {a }l−1.
ui0 i=0 vi0 i=0
10