A central problem of linear algebra is solving linear systems. Regarding
linear systems as equations over general semirings (V,otimes,oplus,0,1) instead
of rings or fields makes traditional approaches impossible. Earlier work shows
that the solution space X(A;w) of the linear system Av = w over the class of
semirings called join-blank algebras is a union of closed intervals (in the
product order) with a common terminal point. In the smaller class of max-blank
algebras, the additional hypothesis that the solution spaces of the 1x1 systems
Av = w are closed intervals implies that X(A;w) is a finite union of closed
intervals. We examine the general case, proving that without this additional
hypothesis, we can still make X(A;w) into a finite union of quasi-intervals

Jananthan, Hayden

Kepner, Jeremy

Kim, Suna

English

arXiv

Hayden Jananthan

Suna Kim

Jeremy Kepner

Crossref

Linear systems over join-blank algebras

A central problem of linear algebra is solving linear systems. Regarding linear systems as equations over general semirings (V, ⊕, ⊗, 0,1) instead of rings or fields makes traditional approaches impossible. Earlier work shows that the solution space X(A, w) of the linear system Av = w over the class of semirings called join-blank algebras is a union of closed intervals (in the product order) with a common terminal point. In the smaller class of max-blank algebras, the additional hypothesis that the solution spaces of the 1 × 1 systems A ⊗ v = w are closed intervals implies that X(A, w) is a finite union of closed intervals. We examine the general case, proving that without this additional hypothesis, we can still make X(A, w) into a finite union of quasi-intervals

Caltech Authors - Main

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Linear Systems over Join-Blank Algebras
Hayden Jananthan
Mathematics Department
Vanderbilt University
Nashville, Tennessee
hayden.r.jananthan@vanderbilt.edu
Suna Kim
Mathematics Department
California Institute of Technology
Pasadena, California
skim3@caltech.edu
Jeremy Kepner
Lincoln Laboratory Supercomputing Center
Massachusetts Institute of Technology
Lexington, Massachusetts
kepner@ll.mit.edu
Abstract—A central problem of linear algebra is solving linear
systems. Regarding linear systems as equations over general
semirings (V,⊕,⊗, 0, 1) instead of rings or fields makes tra-
ditional approaches impossible. Earlier work shows that the
solution space X(A,w) of the linear system Av = w over the
class of semirings called join-blank algebras is a union of closed
intervals (in the product order) with a common terminal point.
In the smaller class of max-blank algebras, the additional
hypothesis that the solution spaces of the 1×1 systems A⊗v = w
are closed intervals implies that X(A,w) is a finite union of
closed intervals. We examine the general case, proving that
without this additional hypothesis, we can still make X(A,w)
into a finite union of quasi-intervals.
Keywords-linear algebra, matrices, lattices, linear systems
I. INTRODUCTION
Linear algebra is a cornerstone of modern computation. In
particular, one approach to solving a problem in application
is reducing it to a linear-algebraic problem, such as carrying
out a matrix multiplication, solving a linear system, or finding
eigenvalues and eigenvectors.
As data become more varied, new mathematics must be
developed to handle linear algebra over more general algebraic
structures. For example, the need for a variety of data types
to be supported exists in the context of polystore databases
[1] and prompted the creation of the Dynamic Distributed
Dimensional Data Model (D4M) [2] which provides a linear
algebraic interface to graphs stored in NoSQL [3], [4], SQL
[5], [6], and NewSQL [7].
One of the most general algebraic structures over which
linear algebra makes sense is a semiring. Semirings include
many algebraic structures that we often encounter – in particu-
lar, all rings and fields are semirings. Among the most studied
semirings which are not rings include the max-plus algebra
R ∪ {∞,−∞}, which forms a semiring with addition max
and multiplication +, and the max-min algebra R∪{∞,−∞},
which forms a semiring with addition max and multiplication
min [8], [9].
In fact, mathematicians and scientist have found numerous
applications of max-plus algebra; it is widely used fields like
in performance evaluation of manufacturing systems, discrete
event system theory, Markov decision processes, and even in
language theory [10].
Note that a semiring generalizes the notion of a ring by
dropping the necessity of additive inverses existing. One
method of dealing with the loss of subtraction is to make use
of non-algebraic properties, particularly strong order-theoretic
properties [11]. Thus we focus on the semirings that are
induced from ordered sets (as in max-plus algebra) and utilize
those properties to characterize the solution set.
II. DEFINITIONS
The most basic object of study is that of a semiring.
Definition II.1 (Semiring). [12], [13] A semiring is a quintu-
ple (V,⊕,⊗, 0, 1) consisting of
1) an underlying set V ,
2) two binary operations ⊕ (addition) and ⊗ (multiplica-
tion) on V , and
3) two elements 0 and 1 of V
such that
1) ⊕ is associative, commutative, and has identity element
0,
2) ⊗ is associative and has identity element 1,
3) ⊗ distributes over ⊕, and
4) 0 is a multiplicative annihilator.
Matrices and their operations can be defined over general
semirings, in a similar way it is over fields like R or C.
Definition II.2 (Matrices). An m× n matrix over a semiring
V is a map
A : {1, . . . ,m} × {1, . . . , n} → V
If A and B are two m×n matrices, their sum is the m×n
matrix A⊕B defined by
(A⊕B)(i, j) = A(i, j)⊕B(i, j)
If A is an m × n matrix and B is an n × p matrix, their
product is the m× p matrix AB defined by
AB(i, j) =
n⊕
k=1
A(i, k)⊗B(k, j)
Elements of the Cartesian product V n are identified with
n× 1 matrices over V .
Definition II.3 (Linear Systems). An m × n linear system
over V is an equation of the form Av = w where A is a
fixed m × n matrix, w is a fixed m × 1 matrix, and v is a
variable n× 1 matrix.
The solution space X(A,w) of a linear system Av = w
is the set
X(A,w) = {v | Av = w}
One nice class of semirings which is diametrically opposite
of the notion of a ring is that of join-blank algebras, which
make explicit and extended use of an underlying order by
requiring that the underlying set be a complete lattice, the
addition operation be binary supremum, and the multiplication
operation satisfy an “infinite-distributivity” law.
Definition II.4 (Complete Lattice). A pair (V,≤) of a set V
and a binary relation ≤ on V is a complete lattice if
1) ≤ is reflexive, antistymmetric, and transitive,
2) for any subset U ⊂ V there exists a least element
∨
U
greater than or equal to every element of U , called the
join or supremum of U , and
3) for any subset U ⊂ V there exists a greatest element∧
U less than or equal to every element of U , called
the meet or infimum of U .
In the case of a two element set {u, v}, the join of {u, v}
is denoted ∨
{u, v} = u ∨ v
and its meet is denoted∧
{u, v} = u ∧ v
These binary operations ∨ and ∧ are called join and meet,
respectively. Semirings in which the underlying set and the
operations have order-theoretic properties with respect to a
fixed partial order allows for order-theoretic tools to be applied
to the construction of solution sets in terms of intervals.
Definition II.5 (Join-Blank Algebra). A join-blank algebra is
a semiring (V,∨,⊗,−∞, 1) where
1) V is a complete lattice with respect to some fixed order,
2) ∨ is the join with respect to that order,
3) −∞ is the minimum element of V with respect to that
order, and
4) for any subset U ⊂ V and element v ∈ V
v ∧
∨
U =
∨
{v ∧ u | u ∈ U}
The max-plus algebra (R∪{−∞,∞},max,+,−∞, 0) and
the max-min algebra (R∪ {−∞,∞},max,min,−∞,∞) are
join-blank algebras. Power set algebras (P(S),∪,∩, ∅, S) and
more generally Heyting algebras form join-blank algebras.
[14]
The order-theoretic properties of a join-blank algebra V can
be extended to the Cartesian product V n.
Definition II.6 (Product Order). Suppose V is ordered by ≤.
The product order ≤ on V n is defined by
v ≤ w if and only if v(i) ≤ w(i) for all i
III. JOIN-BLANK STRUCTURE THEOREM
The order-theoretic properties of a join-blank algebra V
extend to order-theoretic properties of V n.
Proposition III.1. [14] If V is a complete lattice, then V n
is a complete lattice. Moreover, if U ⊂ V n then
∨
U exists if
and only if
∨
{v(i) | v ∈ U} exists for each i, in which case(∨
U
)
(i) =
∨
{v(i) | v ∈ U}
The compatibility of ∨ and ⊗ with the order contribute to
order-theoretic properties of X(A,w):
Proposition III.2. [14] Suppose Av = w is a linear system
over a join-blank algebra V . (a)
1) X(A,w) is closed under taking joins of non-empty
subsets.
2) X(A,w) is convex, so if v1 ≤ v2 ≤ v3 and v1,v3 ∈
X(A,w), then v2 ∈ X(A,w).
This implies the following structure of X(A,w) as a union
of closed intervals with a common terminal point. Recall that
a closed interval is defined as
[x,y] = {z | x ≤ z ≤ y}
Theorem III.3 (Join-Blank Structure Theorem). [14] Suppose
Av = w is a linear system over a join-blank algebra V . Then
X(A,w) is of the form
X(A,w) =
⋃
v∈U
[v,x]
for some U ⊂ X(A,w) and a fixed x.
This structure allows for the problem of finding the solution
space X(A,w) to be reduced to finding the solution spaces
X(A(i, :),w(i))
to the single-equation linear systems
A(i, :)v = w(i)
Intersecting each solution set will give us the complete solution
set for the original linear system, as the intersection will satisfy
all equations.
IV. MAX-BLANK STRUCTURE THEOREM
When V is totally-ordered and hence ∨ = max, we call V
a max-blank algebra.
In nice cases, the solution space of a linear system over
a max-blank algebra is a finite union of closed intervals.
However, it is not always the case. Consider the following
system in max-blank algebra:
[
∞
] [
v
]
=
[
∞
]
The solution space is given by (−∞,∞], which cannot be
written as finite unions of intervals. In the 1 × 1 case, the
solution set can be represented with finitely many closed
intervals if
X(A,w) = {v | A⊗ v = w}
is a closed interval.
Theorem IV.1 (Max-Blank Structure Theorem for Closed
Intervals). [14] Suppose Av = w is a linear system over
a max-blank algebra such that for every i, j the set
{v | A(i, j)⊗w(i)}
is a closed interval. Then X(A,w) is a finite union of closed
intervals.
The crucial step in the proof of Theorem IV.1 is that a
Cartesian product of closed intervals in V is a closed interval
in V n in the product order.
Proposition III.2 shows that the only other form that
{v | A(i, j)⊗w(i)}
can take on is a half-open interval which is open on the left.
The Cartesian product of arbitrary intervals in V need to be
an interval in V n in the product order.
This motivates a slightly more general basic object than
intervals.
Definition IV.2 (Quasi-interval). Suppose I1, . . . , In are in-
tervals in V . Then define
A[(p,q)]B = I1 × · · · × In
where Ik has endpoints p(k) and q(k), with exclusion of p(k)
when k ∈ A and exclusion of q(k) when k ∈ B.
Lemma IV.3. Suppose V is totally ordered. Suppose A,B ⊂
{1, . . . , n} and p,q, r, s ∈ V n. Then
A[(p,q] ∩ B[(r, s] = C [(p ∨ r,q ∧ s]
where
C = {i ∈ A \B | p(i) ≥ r(i)}
∪ {j ∈ B \A | p(j) ≤ r(j)}
∪ A ∩B
Proof: Let
A[(p,q] = I1 × · · · × In
and
B[(r, s] = J1 × · · · × Jn
Then
A[(p,q] ∩ B[(r, s] = (I1 × · · · × In) ∩ (J1 × · · · × Jn)
= (I1 ∩ J1)× · · · × (In ∩ Jn)
Since the intersection of intervals is also interval, each of Ik∩
Jk is an interval with end-points p(k)∨ r(k) and q(k)∧ s(k)
with exclusion of the first end-point p(k)∨r(k) exactly when
either (i)
1) k ∈ A \B and p(k) ≥ r(k), or
2) k ∈ B \A and p(k) ≤ r(k), or
3) k ∈ A ∩B.
and inclusion of the second end-point q(k) ∧ s(k). Hence
A[(p,q] ∩ B[(r, s] = C [(p,q]
where
C = {i ∈ A \B | p(i) ≥ r(i)}
∪ {j ∈ B \A | p(j) ≤ r(j)}
∪ (A ∩B)
Using this new notation, we show that linear systems in
max-blank algebra have solution set that can be written as
union of finite quasi-interval.
Theorem IV.4 (Max-blank Structure Theorem). Suppose A is
an n×m matrix and w an element of V n. Let Ui, Ui,1, and
Ui,2 be the sets of j ∈ {1, . . . ,m} such that
X(A(i, j),w(i)) = {v ∈ V | A(i, j)⊗ v = w(i)}
is non-empty, non-empty and not a closed interval, and
non-empty and a closed interval, respectively. When j ∈
Ui,1, let X(A(i, j),w(i)) = (p
i
j , q
i
j ]. For j ∈ Ui,2, let
X(A(i, j),w(i)) = [pij , q
i
j ]. Lastly, for j /∈ Ui let q
i
j be the
largest element such that A(i, j)⊗ qij ≤ w(i).
Let pi,j′ be defined by
pi,j′(j) =
{
pij′ j = j
′
−∞ otherwise
and qi be defined by qi(j) = q
i
j . Then
X(A,w) =
⋃
j′i,k∈Ui,k
for 1 ≤ i ≤ n
and k ∈ {1, 2}
Jj
[( ∨
1≤i≤n,k∈{1,2}
pi,j′
i,k
,
∧
1≤i≤n
qi
]
where j = (j′i,k)1≤i≤n,k∈{1,2} and ℓ ∈ Jj if and only if
max
1≤i≤n
pi,ji,2 ≤ max
1≤i≤n
pi,ji,1
Proof: The proof will consist of finding the solution
space of maxj∈{1,...,m} (A(i, j)⊗ v(j)) = w(i) for each
i ∈ {1, . . . , n} and showing that it is the union of intervals
with a common (inclusive) terminal point. Taking the inter-
section of these solution spaces is X(A,w).
X([v], [u]) is convex with a inclusive terminal point. Since
V is a complete lattice, it follows that
X(A(i, j),w(i)) = [pij , q
i
j ]) or X(A(i, j),w(i)) = (p
i
j , q
i
j ]
for some pij and q
i
j , assuming that X(A(i, j),w(i)) is non-
empty.
Let define Ui, Ui,1, and Ui,2 as in the proposition. For
j /∈ Ui, let qij be the largest element such that A(i, j)⊗ q
i
j ≤
w(i). Such an element exists since −∞ ⊗ v = −∞ and
multiplication by a fixed element is a monotonic map.
The solution space now can be written down in terms of
the elements pij and q
i
j . A given v is in the solution set if and
only if there exists a j′ ∈ {1, . . . ,m} such that
A(i, j′)⊗ v(j′) = w(i)
and for all j ∈ {1, . . . ,m} it is true that
A(i, j)⊗ v(j) ≤ w(i)
The first condition is that v(j′) ∈ f−1
A(i,j′)(w(i)) and the sec-
ond condition is that v(j) ∈ [−∞, qij ] because multiplication
by a fixed element is a monotonic function. Then the solution
space can be written as
⋃
j′∈Ui

X(A(i, j′),w(i))× ∏
j∈{1,...,m},j 6=j′
[−∞, qij ]


=
⋃
j′∈Ui,1

(pij′ , qij′ ]× ∏
j∈{1,...,m},j 6=j′
[−∞, qij ]


∪
⋃
j′∈Ui,2

[pij′ , qij′ ]× ∏
j∈{1,...,m},j 6=j′
[−∞, qij]


Taking pi,j′(j) and qi(j) as defined before, we get
[pij′ , q
i
j′ ]×
∏
j∈{1,...,m},j 6=j′
[−∞, qij] = [pi,j′ ,qi]
and
(pij′ , q
i
j′ ]×
∏
j∈{1,...,m},j 6=j′
[−∞, qij] = j′ [(pi,j′ ,qi]
Let
Ii,j =
{
{j} if j ∈ Ui,1
∅ otherwise
Using this, we can change the order of intersection with
union to get
X(A,w) =
n⋂
i=1

 ⋃
j′∈Ui,1
j′ [(pi,j′ ,qi] ∪
⋃
j′∈Ui,2
[pi,j′ ,qi]


=
⋃
j′i,k∈Ui,k
for 1 ≤ i ≤ n
and k ∈ {1, 2}
⋂
1≤i≤n,k∈{1,2}
Ii,j′
i,k
[(pi,j′
i,k
,qi]
Finally, since the intersection of quasi-intervals can be
represented as a quasi-interval as well (Lemma IV.3), we arrive
at
X(A,w) =
⋃
j′i,k∈Ui,k
for 1 ≤ i ≤ n
and k ∈ {1, 2}
Jj
[( ∨
1≤i≤n,k∈{1,2}
pi,j′
i,k
,
∧
1≤i≤n
qi
]
where j = (j′i,k)1≤i≤n,k∈{1,2} and ℓ ∈ Jj ⊂ {1, . . . , n} if and
only if
max
1≤i≤n
pi,j′
i,2
(ℓ) ≤ max
1≤i≤n
pi,j′
i,1
(ℓ)
V. FURTHER RESEARCH
The notion of a quasi-interval allows the structure of the
solution space of a linear system over a max-blank algebra to
be written as a finite union of quasi-intervals. This naturally
leads to the question of how this notion can be used to express
other solution spaces in similarly nice ways.
While the maximum solution of a linear system is known
in many cases, particularly for Heyting algebras (a join-blank
algebra in which ⊗ is the meet) and max-blank algebras, the
entire structure is not known for arbitrary Heyting algebras.
Also worth investigating is how crucial each of the proper-
ties a join-blank algebra satisfies are to Theorem III.3.
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Linear Systems over Join-Blank Algebras

Linear Systems over Join-Blank Algebras

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Caltech Authors - Main

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