upper triangular matrices over semirings · 2020-06-26 · upper triangular matrices over semirings...

Upper triangular matrices oversemiringsA study of their factorization

Nicholas R. Baeth

Franklin & Marshall College

25 06 2020

Motivation

Motivating example

Example (Factorization in T2(Z)•)

We can factor A =

(6 20 5

)∈ T2(Z)• as:

A =

(1 00 5

)(1 10 1

)2(2 00 1

)(3 00 1

)Considering determinants, this and all other factorizations of Ainvolve exactly 3 irreducibles.

Theorem (Bachman-B-Gossell [BBG14])ϕ :

(a b0 c

)7→ (a, c) is a weak transfer homomorphism from

T2(Z)• to (Z\0)n.

Factorization in T2(Z)• is governed entirely by (Z, ·).

1 25

Motivating example


We can factor A =

(6 20 5

)∈ T2(Z)• as:

A =

(1 00 5

)(1 10 1

)2(2 00 1

)(3 00 1

)

Considering determinants, this and all other factorizations of Ainvolve exactly 3 irreducibles.


(a b0 c


T2(Z)• to (Z\0)n.


1 25

Motivating example


We can factor A =

(6 20 5

)∈ T2(Z)• as:

A =

(1 00 5

)(1 10 1

)2(2 00 1

)(3 00 1



(a b0 c


T2(Z)• to (Z\0)n.


1 25

Motivating example


We can factor A =

(6 20 5

)∈ T2(Z)• as:

A =

(1 00 5

)(1 10 1

)2(2 00 1

)(3 00 1



(a b0 c


T2(Z)• to (Z\0)n.

Factorization in T2(Z)• is governed entirely by (Z, ·).1 25

From Z to N0

Example (Factorization in T2(N0)•)

We can factor A =

(6 20 5

)∈ T2(N0)• as:

A =

(1 00 5

)(1 10 1

)2(2 00 1

)(3 00 1

)The identity is the only unit of T2(N0)•, and this is a product ofexactly 5 irreducible elements.

Moreover, there is no weak transfer homomorphism from T2(N0)•

to any commutative monoid.

Goals:Generalize N0 in Z to semirings of atomic domains.Study factorization in Tn(S)• where S is such a semiring

2 25

From Z to N0


We can factor A =

(6 20 5

)∈ T2(N0)• as:

A =

(1 00 5

)(1 10 1

)2(2 00 1

)(3 00 1

)

The identity is the only unit of T2(N0)•, and this is a product ofexactly 5 irreducible elements.




2 25

From Z to N0


We can factor A =

(6 20 5

)∈ T2(N0)• as:

A =

(1 00 5

)(1 10 1

)2(2 00 1

)(3 00 1

)The identity is the only unit of T2(N0)•, and this is a product ofexactly 5 irreducible elements.




2 25

Definitions and Terminology

Monoids

DefinitionA monoid semigroup with identity; that is, a set S closed withrespect to some (not necessarily commutative) associative,cancellative (a ∗ b = a ∗ c⇒ b = c) operation ∗.An element u ∈ S, a monoid with identity 1, is a unit if thereis v ∈ S with u ∗ v = 1 = v ∗ u.A monoid is reduced if it has only one invertible element.A nonunit a ∈ S is an atom if a = x ∗ y in S implies x or y is aunit.A factorization of x ∈ S is a representation x = a1 ∗ · · · ∗ anwith each ai an atom of S.

3 25

DefinitionLet (S, ∗) be a monoid.

S is atomic if every nonunit can be written as a product ofatoms.An ideal in S is a nonempty subset I with S ∗ I, I ∗ S ⊆ I. Theideal I is principal if I = Sa = aS for a ∈ S.The monoid S satisfies the ACCP provided every ascendingchain of principal ideals stabilizes.If x = a1 · · ·a` with ai atoms of S, ` is a length of x. The set ofall possible lengths is the set of lengths L(x).S is a bounded factorization monoid (BFM) if |L(x)| <∞ forall x ∈ S. S is a half-factorial monoid (HFM) if |L(x)| = 1 for allx ∈ S.S is a finite factorization monoid (FFM) if every x ∈ S hasfinitely many factorizations.

4 25

Theorem (Anderson-Anderson-Zafrullah [AAZ90])For any commutative monoid (or integral domain), the followingimplications always hold. In general, the arrows are notreversible.

HFM

UFM BFM ACCP atomic

FFM

5 25

Generalizing N0 in Z

A triple (S,+, ·) is a semiring if:(S,+) is a commutative monoid with identity element 0.(S, ·) is a semigroup with identity element 1 6= 0.

Multiplication distributes over addition.

0 · x = 0 for all x ∈ S.

DefinitionA semiring S is an information semialgebra if (S,+) is a reducedmonoid and (S \ 0, ·) is a commutative monoid. S is reduced if(S, ·) is reduced.

Throughout, (S,+, ·) will denote a reduced informationsemialgebra where both (S,+) and (S, ·) are reducedcommutative cancellative monoids.

6 25

Information Semialgebras

Example (Information semialgebras)

1. (N0,+, ·)2. N0[

√d] where d is a positive integer.

3. N0[x], a semiring of polynomials with nonnegative integercoecients

4. Any numerical monoid S; a complement-finite additivesubmonoid of N0.

5. Any Puiseux information semialgebra S, where (S,+) isisomorphic to a Puiseux monoid (an additive submonoid ofthe nonnegative cone of rational numbers ) containing 1.

RemarkFactorizations in N0[

√d], N0[x], and Puiseux monoids studied in

[CCMS09] [CF19], and [CGG20], respectively.7 25

Semialgebra atoms

DefinitionLet (S,+, ·) be a reduced information semialgebra.

We denote by A+(S), the additive atoms of S; a ∈ S\0 suchthat a = b + c ⇒ b = 0 or c = 0.A×(S) denotes the multiplicative atoms; a ∈ S\0, 1 suchthat a = bc ⇒ b = 1 or c = 1.

Lemma (B-Gotti [BS20])For an information semialgebra S, the following are equivalent.

1. (S,+) is atomic and |A+(S)| = 1.2. As monoids, (S,+) ∼= (N0,+).3. As semirings, (S,+, ·) ∼= (N0,+, ·).

8 25

Triangular matrices over infor-mation semialgebras

Basic Results

DefinitionLet S denote a reduced information semialgebra. Then Tn(S)•

denotes the monoid of n× n upper-triangular matrices withnonzero determinant.

Proposition (B-Sampson [BS20])1. If S is a reduced information semialgebra, then Tn(S)• is

reduced, with In the only invertible element.2. If (S,+) and (S, ·) are atomic, then Tn(S)• is atomic and the

atoms of Tn(S)• are the following:I Iij(α) = I + αEij with 1 ≤ i < j ≤ n and α ∈ A+(S).I Iii(a) = I + (a− 1)Eii with 1 ≤ i ≤ n and a ∈ A×(S).

9 25

PropositionIf S is a reduced information semialgebra, T2(S) is reduced.

Proof.

Suppose(

a1 b10 c1

)(a2 b20 c2

)= I.

Then a1a2 = 1 = c1c2. Since (S, ·) is reduced, a1 = a2 = c1 = c2 = 1.

Also, 0 = a1b2 + b1c2 = b2 + b1. Since (S,+) is reduced,b1 = b2 = 0 S.

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PropositionIf S is a reduced information semialgebra, the atoms of T2(S) are:

( a 00 1 ) and ( 1 0

0 a ) with a ∈ A×(S)

( 1 α0 1 ) with α ∈ A+(S)

Proof.For any a,b, c ∈ S, note that

A =

(a b0 c

)=

(1 00 c

)(1 b0 1

)(a 00 1

)With c = c1 · · · cm, b = b1 + · · ·+ bl, and a = a1 · · ·ak with eachai, cj ∈ A×(S) and each bi ∈ A+(S), we can further factor A as:(

1 00 c1

)· · ·(

1 00 cm

)(1 b10 1

)· · ·(

1 bl0 1

)(a1 00 1

)· · ·(

ak 00 1

)

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PropositionIf S is a reduced information semialgebra, the atoms of T2(S) are:

( a 00 1 ) and ( 1 0

0 a ) with a ∈ A×(S)

( 1 α0 1 ) with α ∈ A+(S)

Proof.For any a,b, c ∈ S, note that

A =

(a b0 c

)=

(1 00 c

)(1 b0 1

)(a 00 1

)With c = c1 · · · cm, b = b1 + · · ·+ bl, and a = a1 · · ·ak with eachai, cj ∈ A×(S) and each bi ∈ A+(S), we can further factor A as:(

1 00 c1

)· · ·(

1 00 cm

)(1 b10 1

)· · ·(

1 bl0 1

)(a1 00 1

)· · ·(

ak 00 1

)11 25

DefinitionA monoid is transfer-Krull provided there is a (weak) transferhomomorphism to a commutative Krull monoid.

Proposition (B-Sampson [BS20])For n ≥ 2 and S a reduced information semialgebra, Tn(S)• is nottransfer-Krull

Proof.Suppose ϕ : T2(S)• → M is a (weak) transfer homomorphism to acommutative cancellative monoid. Then, with α ∈ A+(S),

( 2 00 1 )( 1 α

0 1 ) = ( 1 α0 1 )2( 2 0

0 1 )⇒ ϕ( 2 00 1 )ϕ( 1 α

0 1 ) = ϕ( 2 00 1 )ϕ( 1 α

0 1 )2.

Thus so ϕ( 1 α0 1 ) = 1M, a contradiction.

12 25

Factorization in Tn(S)•

Divisor-closed submonoids

DefinitionA submonoid S′ of a monoid S is divisor-closed provided thatwhenever x ∈ S and y ∈ S′ and x divides y, x ∈ S′.

Theorem (B-Gotti [BG])Let S be a reduced information semialgebra and let n ≥ 2. Thefollowing are divisor-closed submonoids of Tn(S)•.

Un(S) = A = (aij) ∈ Tn(S)• : aii = 1∀ iI + (s− 1)Eii : s ∈ S• ∼= (S•, ·)I + sEij : s ∈ S ∼= (S,+)

I + sEij : s ∈ S is also divisor-closed in Un(S).

13 25

Reduction to the 2× 2 case

Proposition (B-Sampson [BS20])Let S be a reduced information semialgebra, set n > 2, and defineϕ : T2(S)• → Tn(S)• by

ϕ :

(a b0 c

)7→

a b0 c 00 In−2

.

Then T2(S)• ∼= ϕ(T2(S)•), a divisor-closed submonoid of Tn(S)•.

Consequently, many proofs can be reduced immediately to the2× 2 case.

14 25

Half-factoriality

Proposition (B-Sampson [BS20])Tn(S)• is not an HFM for any reduced information semialgebra Sand any n ≥ 2.

Proof.(S,+) ∼= I + sE12 : s ∈ S is divisor closed in Tn(S)• so if (S,+) isnot atomic, neither is Tn(S)•.

If (S,+) is atomic, then 1 ∈ A+(S). Otherwise, 1 = x + y for somex, y ∈ S• implies s = s(x + y) = sx + sy ∈ S• + S• for all s ∈ S•.

A :=

(1 00 m

)(1 10 1

)m=

(1 10 1

)(1 00 m

).

Now ` ∈ LT2 (( 1 00 m )) implies `+ 1, `+ m ∈ LT2(A).

15 25

Bounded Factorizations

ExampleClearly (N0,+) and (N, ·) are BFMs. For A ∈ Tn(N0)•,max L(A) = Ω(det(A)) + Σ(A) <∞ and so Tn(N0)• is a BFM.

ExampleLet S be the additive submonoid of Q≥0 generated by the setA = (2/3)n : n ∈ N0. S is a redued information semialgebra.(S,+) is an atomic monoid with set of atoms A.For every n ∈ N, the 3 = 2

( 23)n

+∑n

i=0( 2

3)i and so (S,+) is not a

BFM. Moreover, T3(S)• is not a BFM since( 1 0 30 1 00 0 1

)=

(1 0(

23

)n

0 1 00 0 1

)2( 1 0 10 1 00 0 1

)( 1 0 23

0 1 00 0 1

)· · ·(

1 0(

23

)n

0 1 00 0 1

)for every n ∈ N.

16 25



ExampleLet S be the additive submonoid of Q≥0 generated by the setA = (2/3)n : n ∈ N0. S is a redued information semialgebra.

(S,+) is an atomic monoid with set of atoms A.For every n ∈ N, the 3 = 2

( 23)n

+∑n

i=0( 2



)=

(1 0(

23

)n

0 1 00 0 1

)2( 1 0 10 1 00 0 1

)( 1 0 23

0 1 00 0 1

)· · ·(

1 0(

23

)n

0 1 00 0 1

)for every n ∈ N.

16 25



ExampleLet S be the additive submonoid of Q≥0 generated by the setA = (2/3)n : n ∈ N0. S is a redued information semialgebra.(S,+) is an atomic monoid with set of atoms A.

For every n ∈ N, the 3 = 2( 2

3)n

+∑n

i=0( 2



)=

(1 0(

23

)n

0 1 00 0 1

)2( 1 0 10 1 00 0 1

)( 1 0 23

0 1 00 0 1

)· · ·(

1 0(

23

)n

0 1 00 0 1

)for every n ∈ N.

16 25



ExampleLet S be the additive submonoid of Q≥0 generated by the setA = (2/3)n : n ∈ N0. S is a redued information semialgebra.(S,+) is an atomic monoid with set of atoms A.For every n ∈ N, the 3 = 2

( 23)n

+∑n

i=0( 2



)=

(1 0(

23

)n

0 1 00 0 1

)2( 1 0 10 1 00 0 1

)( 1 0 23

0 1 00 0 1

)· · ·(

1 0(

23

)n

0 1 00 0 1

)for every n ∈ N.

16 25

Finite Factorizations

ExampleClearly (N0,+) and (N, ·) are FFMs. For A ∈ Tn(N0)•, there are atmost Ω(det(A)) + (n2 − n)/2 atoms that divide it, each appearingat most finitely many times, and so Tn(N0)• is a FFM.

ExampleThe subsemiring S = 0, 1 ∪Q≥2 of Q is a reduced informationsemialgebra with A×(S) = [2, 4) ∩Q.

In particular, for all n ∈ N, an = 3nn+1 and bn = 3(n+1)

n aremultiplicative atoms of S.

T2(S)• is not an FFM since ( 9 00 1 ) = ( an 0

0 1 )(

bn 00 1)

for all n ∈ N.

17 25

Two operations

RemarkIf D is an atomic domain, the arithmetic of Tn(D)• is completelydetermiend by that of (D•, ·).

DefinitionIf S is a reduced information semialgebra and ∈ FFM, BFM, ACCP, atomic, then we say that S is a bi-provided that both its additive monoid (S,+) and multiplicativemonoid (S•, ·) are .

As we have observed, the arithmetic of Tn(S)• depends on thearithmetic of both (S,+) and (S, ·).

18 25

Factorization in Tn(S)•

Theorem (B-Gotti [BG])Let S be a reduced information semialgebra. For n ≥ 2, eachimplication in the following diagram holds.

Tn(S)• is FFM Tn(S)• is BFM Tn(S)• satisfies ACCP Tn(S)• is atomic

S is bi-FFM S is bi-BFM S satisfies bi-ACCP S is bi-atomic

(S,+) is FFM (S,+) is BFM (S,+) satisfies ACCP (S,+) is atomic

Un(S) is FFM Un(S) is BFM Un(S) satisfies ACCP Un(S) is atomic

Moreover, none of the horizontal implications is, in general,reversible. Finally, Tn(S)• is never an HFM when n ≥ 2.

19 25

Arithmetical results:a focus on Tn(N0)

Atoms

DefinitionAn atom is almost prime-like if whenever it appears in onefactorization of an element it appears in all factorizations ofthat element.

An almost prime-like element is prime-like if for each fixedelement A, it always appears with the same multiplicity inany fatorization of A.

An atom A is absolutely irreducible if no other atoms divideAn for any n ∈ N.

20 25

Theorem(B-Sampson-Chen-Liu-Heilbrunn-Young [BS20, BC+])Consider the monoid Tn(N0)• for some n ≥ 2.

The atoms are:I Iij(1) = I + Eij with 1 ≤ i < j ≤ n.I Iii(p) = I + (p− 1)Eii with 1 ≤ i ≤ n and p prime in N.

Atoms of the form Iii(p) are prime-like.Atoms of the form Iij(1) are almost prime-like but notprime-like if j = i + 1.Atoms of the form Iij(1) are not almost prime-like if j > i + 1.All atoms of Tn(N0)• are absolutely irreducible.

21 25

Delta set

DefinitionIf L(A) = `1 < `2 < · · · , ∆(A) = `i+1 − `i : i ≥ 1.

Proposition (B-Sampson [BS20])∆(Tn(N0)•) = ∪∆(A) = N for all n ≥ 2.

Proof.Fix k ∈ N and choose a prime p > 2k + 2. Set M =

( p 2p−2k−20 2

).

Atoms dividing M: A = ( 1 00 2 ), B = ( 1 1

0 1 ), C =( p 0

0 1). Factorizations:

AB2p−2k−2C = BAB2p−2k−4 = · · · = Bp−k−1AC and

ABp−2k−2CB = BABp−2k−4CB = · · · = Bp−2k−3

2 ABCB

L(M) = [(p + 5)/2− k,p− 2k + 1] ∪ [p− k + 1, 2p− 2k] .

22 25

Delta set




( p 2p−2k−20 2

).


0 1 ), C =( p 0

0 1).

Factorizations:



2 ABCB

L(M) = [(p + 5)/2− k,p− 2k + 1] ∪ [p− k + 1, 2p− 2k] .

22 25

Delta set




( p 2p−2k−20 2

).


0 1 ), C =( p 0

0 1). Factorizations:



2 ABCB

L(M) = [(p + 5)/2− k,p− 2k + 1] ∪ [p− k + 1, 2p− 2k] .

22 25

Elasticity

Proposition (B-Sampson [BS20])Tn(N0)• has full infinite elasticity: For each q ∈ Q≥1 q = t/s withA1 · · ·As = B1 · · ·Bt two products of atoms.

Proof. (p 00 1

)(1 10 1

)=

(1 10 1

)p(p 00 1

)for any prime p and so ρ(Tn(N0)• =∞. For r

s ∈ Q≥1, write rs = k+n

k+1for k,n ∈ N. If Ω(n) ≥ k, set

A =

(2k−Ω(n)n n

0 1

).

Then ρ(A) = k+nk+1 . We adjust A in case Ω(n) < k.

23 25

Elasticity


Proof. (p 00 1

)(1 10 1

)=

(1 10 1

)p(p 00 1

)for any prime p and so ρ(Tn(N0)• =∞.

For rs ∈ Q≥1, write r

s = k+nk+1

for k,n ∈ N. If Ω(n) ≥ k, set

A =

(2k−Ω(n)n n

0 1

).


23 25

Elasticity


Proof. (p 00 1

)(1 10 1

)=

(1 10 1

)p(p 00 1

)for any prime p and so ρ(Tn(N0)• =∞. For r

s ∈ Q≥1, write rs = k+n

k+1for k,n ∈ N. If Ω(n) ≥ k, set

A =

(2k−Ω(n)n n

0 1

).


23 25

Unions of sets of lengths

DefinitionFor a monoid S and k ≥ 2, the union of sets of lengths of Scontaining k is Uk(S) = n : a1 · · ·ak = b1 · · ·bn,ai,bj ∈ A(S).

Proposition (B-Chen-Liu-Heilbrunn-Young [BC+])Let n > 1. Then:

U2(Tn(N0)•) = 2 ∪ p + 1 : p prime and

Uk(Tn(N0)•) = N≥3 ∪ S for k ≥ 3 where

S =

2, if k ∈ 2 ∪ p + 1 : p prime∅, otherwise

24 25

In summary

Factorization in Tn(S)• is quite dierent from that in Tn(D)•.

Both the multiplicative and additive structures of S influencefactorizaiton in Tn(S)•.

Factorization in Tn(S)• is highly non-unique, even with nsmall and with (S,+) and (S, ·) well-behaved commutativemonoids.

25 / 25

Thanks for listening!

References

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Nicholas R. Baeth, , Santure Chen, Gregory Heilbrunn, Liu Peter, and MitchellYoung, The arithmetic of matrices with nonnegative integer entries.

Nicholas R. Baeth and Felix Gotti, Factorizations in upper triangular matrices overinformation semialgebras, to appear in J. Algebra.

Nicholas R. Baeth and Rylan Sampson, Upper triangular matrices over informationalgebras, Linear Algebra Appl. 587 (2020), 334–357. MR 4036676

Patrick Cesarz, S. T. Chapman, Stephen McAdam, and George J. Schaeffer, Elasticproperties of some semirings defined by positive systems, Commutative algebra andits applications, Walter de Gruyter, Berlin, 2009, pp. 89–101. MR 2606280

Federico Campanini and Alberto Facchini, Factorizations of polynomials withintegral non-negative coefficients, Semigroup Forum 99 (2019), no. 2, 317–332. MR4026385Scott T. Chapman, Felix Gotti, and Marly Gotti, Factorization invariants of Puiseuxmonoids generated by geometric sequences, Comm. Algebra 48 (2020), no. 1,380–396. MR 4060036

upper triangular matrices over semirings · 2020-06-26 · upper triangular matrices over semirings...

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