Identities in upper triangular tropical matrix semigroups ...varg1/MarianneYork17.pdf · Semigroup...

Identities in upper triangular tropicalmatrix semigroups and the bicyclic

monoid

Marianne Johnson(Joint work with Laure Daviaud and Mark Kambites)

York Semigroup Seminar, May 2017

Semigroup identities and varieties of semigroups

Let Σ be an alphabet, and consider two words w, v ∈ Σ+. Saythat w = v is a semigroup identity for S if ϕ(w) = ϕ(v) forall semigroup morphisms ϕ : Σ+ → S.

A variety of semigroups is a class of semigroups defined bysatisfaction of a particular set of identities.

Birkhoff’s theorem: A class of semigroups is a variety ifand only if it is closed under taking subsemigroups, directproducts and homomorphic images.

By the variety generated by a semigroup S we mean thevariety defined by the set of all identities satisfied by S.

General Questions

Given a semigroup S we may ask...

I Does S satisfy a non-trivial semigroup identity?

Golubchik and Mikhalev, 1978: For n > 1, K an infinitefield, Mn(K) does not satisfy a non-trivial semigroup identity.

I Can we “describe” the identities satisfied by S?This can be difficult...

There are finite semigroups which are not finitely based!Perkins, 1969: Six element example using 0-1 matrices.Sapir/Volkov, 1980s: Mn(K) where n > 1 and K finite.

I Does S generate the same semigroup variety assome other interesting/well-studied semigroup?

General Questions

I Can we “describe” the identities satisfied by S?

This can be difficult...

General Questions

The basics of tropical arithmetic

Let T denote the tropical semiring (R ∪ {−∞},⊕,⊗), where⊕ and ⊗ denote two binary operations defined by:x⊕ y := max(x, y), x⊗ y := x+ y.

I x⊕ x = x;I x⊗ 0 = 0⊗ x = x.

The semigroup Mn(T) is the set of all n× n tropicalmatrices, with multiplication ⊗ defined in the obvious way:

(2 10 19

)⊗(−1 −1−20 4

(1 5−1 23

)Variants: Let T = R,Q,Z or N0. Can also consider:T̄max := (T ∪ {−∞},⊕,⊗), or Tmax := (T ,⊕,⊗), orT̄min := (T ∪ {+∞},�,⊗), or Tmin := (T ,�,⊗),where x� y = min(x, y).

I x⊕ x = x;I x⊗ 0 = 0⊗ x = x.

(2 10 19

)⊗(−1 −1−20 4

(1 5−1 23

Variants: Let T = R,Q,Z or N0. Can also consider:T̄max := (T ∪ {−∞},⊕,⊗), or Tmax := (T ,⊕,⊗), orT̄min := (T ∪ {+∞},�,⊗), or Tmin := (T ,�,⊗),where x� y = min(x, y).

I x⊕ x = x;I x⊗ 0 = 0⊗ x = x.

(2 10 19

)⊗(−1 −1−20 4

(1 5−1 23

)Variants: Let T = R,Q,Z or N0. Can also consider:T̄max := (T ∪ {−∞},⊕,⊗), or Tmax := (T ,⊕,⊗), orT̄min := (T ∪ {+∞},�,⊗), or Tmin := (T ,�,⊗),where x� y = min(x, y).

Tropical matrix identities

Does Mn(T) satisfy a non-trivial semigroup identity?

Izhakian and Margolis, 2010: For n = 2, yes!

They proved that...

I The bicyclic monoid B embeds into the monoid UT2(T) ofupper triangular tropical matrices.So, any identity satisfied by UT2(T) is also satisfied by B.

I UT2(T) satisfies Adjan’s identity:

abba ab abba = abba ba abba,

thus exhibiting a minimal length identity for UT2(T).

I M2(T) satisfies the identity:

a2b2b2a2 a2b2 a2b2b2a2 = a2b2b2a2 b2a2 a2b2b2a2.

Does Mn(T) satisfy a non-trivial semigroup identity?Izhakian and Margolis, 2010: For n = 2, yes!

They proved that...

Upper triangular tropical matrix identities

I Does UTn(T) satisfy a non-trivial semigroupidentity?

Izhakian 2014, Okninski 2015, Taylor 2016: Yes - differentconstructions.

I Can we “describe” the identities satisfied byUTn(T)?

Daviaud, J, Kambites, 2017: We give a necessary andsufficient condition for an identity to hold in UTn(T).

I Do UT2(T) and B generate the same variety?

Daviaud, J, Kambites, 2017: Yes! Several variants ofUT2(T) and B also generate the same variety.

The case n = 2

Consider morphisms ϕ : Σ+ → UT2(T) of the form:

(F) ϕ(s) =

(xs x′s−∞ 0

), for all s ∈ Σ.

I For all w ∈ Σ+, can see that:

ϕ(w)1,1 =∑t∈Σ

|w|txt, ϕ(w)2,1 = −∞, ϕ(w)2,2 = 0

ϕ(w)1,2 = maxs∈Σ

(maxw=usv

(∑t∈Σ

|u|txt + x′s + 0

= maxs∈Σ

(x′s + maxwi=s

(∑t∈Σ

λwt (i− 1)xt

where |w|t = the number of occurrences of the letter t in w,and λwt (k) = #t’s in the first k letters of w.

The case n = 2

(F) ϕ(s) =

(xs x′s−∞ 0

ϕ(w)1,1 =∑t∈Σ

|w|txt, ϕ(w)2,1 = −∞, ϕ(w)2,2 = 0

(maxw=usv

(∑t∈Σ

|u|txt + x′s + 0

= maxs∈Σ

(x′s + maxwi=s

(∑t∈Σ

λwt (i− 1)xt

The case n = 2

(F) ϕ(s) =

(xs x′s−∞ 0

ϕ(w)1,1 =∑t∈Σ

|w|txt, ϕ(w)2,1 = −∞, ϕ(w)2,2 = 0

(maxw=usv

(∑t∈Σ

|u|txt + x′s + 0

= maxs∈Σ

(x′s + maxwi=s

(∑t∈Σ

λwt (i− 1)xt

The case n = 2

(F) ϕ(s) =

(xs x′s−∞ 0

ϕ(w)1,1 =∑t∈Σ

|w|txt, ϕ(w)2,1 = −∞, ϕ(w)2,2 = 0

(maxw=usv

(∑t∈Σ

|u|txt + x′s + 0

= maxs∈Σ

(x′s + maxwi=s

(∑t∈Σ

λwt (i− 1)xt

The case n = 2

The latter equation can be written tropically as:

ϕ(w)1,2 =⊕s∈Σ

(x′s ⊗

⊕wi=s

⊗t∈Σ

x⊗λwt (i−1)t

)=⊕s∈Σ

(x′s ⊗ fws (x)

Theorem 1: Let w, v ∈ Σ+.The identity w = v holds on UT2(T) if and only iffws (x) = fvs (x) for all s ∈ Σ and all x ∈ RΣ.

I If fws (x) 6= fvs (x) for some s ∈ Σ and x ∈ RΣ, then there isa morphism (F) falsifying the identity in UT2(T).

I If fws (x) = fvs (x) for all s ∈ Σ and all x ∈ RΣ can show thatw and v must have the same content. The result thenfollows from the following (technical) reductions...

The case n = 2

ϕ(w)1,2 =⊕s∈Σ

(x′s ⊗

⊕wi=s

⊗t∈Σ

x⊗λwt (i−1)t

)=⊕s∈Σ

(x′s ⊗ fws (x)

)Theorem 1: Let w, v ∈ Σ+.The identity w = v holds on UT2(T) if and only iffws (x) = fvs (x) for all s ∈ Σ and all x ∈ RΣ.

The case n = 2

ϕ(w)1,2 =⊕s∈Σ

(x′s ⊗

⊕wi=s

⊗t∈Σ

x⊗λwt (i−1)t

)=⊕s∈Σ

(x′s ⊗ fws (x)

The case n = 2

ϕ(w)1,2 =⊕s∈Σ

(x′s ⊗

⊕wi=s

⊗t∈Σ

x⊗λwt (i−1)t

)=⊕s∈Σ

(x′s ⊗ fws (x)

Reductions

UTn(T) and UTn(Rmax) satisfy the same identities.

If ρ : Σ+ → UTn(T) is a morphism falsifying an identity, thenconstruct from this a morphism ρx : Σ+ → UTn(Rmax) whereany −∞ entries on or above the diagonal in ρ(s) are replacedby x� 0 in ρx(s).

Let w, v ∈ Σ+ be words of the same content.If ϕ(w) = ϕ(v) for each morphism ϕ : Σ+ → UTn(Rmax)satisfying ϕ(s)n,n = 0 for all s ∈ Σ, then w = v is an identityin UTn(T).

Follows from the above together with the fact that if ψ and φare morphisms related by ψ(s) = µs ⊗ φ(s) for some µs ∈ Rmax,

φ(w) = φ(v)⇔ ψ(w) = ψ(v).

Reductions

φ(w) = φ(v)⇔ ψ(w) = ψ(v).

Reductions

φ(w) = φ(v)⇔ ψ(w) = ψ(v).

Reductions

φ(w) = φ(v)⇔ ψ(w) = ψ(v).

The case n = 2: Example

By Theorem 1: The identityw := abba ab abba = abba ba abba =: v holds on UT2(T) if andonly if fws (x) = fvs (x) for s = a, b and all x ∈ RΣ, where

fus (x) =⊕ui=s

⊗t∈Σ

x⊗λut (i−1)t

So for example,

fwa (xa, xb) = max(0, xa + 2xb, 2xa + 2xb, 3xa + 3xb, 4xa + 5xb)

= max(0, xa + 2xb, 3xa + 3xb, 4xa + 5xb)

fva (xa, xb) = max(0, xa + 2xb, 2xa + 3xb, 3xa + 3xb, 4xa + 5xb)

fus (x) =⊕ui=s

⊗t∈Σ

x⊗λut (i−1)t

So for example,

fus (x) =⊕ui=s

⊗t∈Σ

x⊗λut (i−1)t

So for example,

fus (x) =⊕ui=s

⊗t∈Σ

x⊗λut (i−1)t

So for example,

The bicyclic monoid

Recall that B = 〈p, q : pq = 1〉 = {qipj : i, j ∈ N0}.

I Izhakian and Margolis: B ↪→ UT2(T) via

qipj 7→(i− j i+ j−∞ j − i

Theorem 2: B and UT2(T) generate the same variety.

I We show that if w 6= v in UT2(T) then we can construct amorphism from Σ+ to the image of B in UT2(T) thatfalsifies the identity.

I Easy to see that this can be done if cont(w) 6= cont(v).

The bicyclic monoid

Recall that B = 〈p, q : pq = 1〉 = {qipj : i, j ∈ N0}.

I Izhakian and Margolis: B ↪→ UT2(T) via

qipj 7→(i− j i+ j−∞ j − i

Theorem 2: B and UT2(T) generate the same variety.

I We show that if w 6= v in UT2(T) then we can construct amorphism from Σ+ to the image of B in UT2(T) thatfalsifies the identity.

I Easy to see that this can be done if cont(w) 6= cont(v).

The bicyclic monoid

Suppose that w 6= v in UT2(T), where cont(w) = cont(v).

I Can assume that fwt (x) > fvt (x) for some t ∈ Σ, x ∈ RΣ.

I The functions are piecewise linear. Inequality aboveholds in some open neighbourhood of x, so can choosex ∈ QΣ.

I All linear expressions involved are homogeneous.Multiplying all entries of x by a positive integer does notchange the inequality, so can choose x ∈ (2Z)Σ.

The bicyclic monoid

I Can assume that fwt (x) > fvt (x) for some t ∈ Σ, x ∈ (2Z)Σ.

I For each s ∈ Σ choose a positive even integer x′s > xs insuch a way that x′s � x′t for s 6= t.

I With these choices, the morphism (F) falsifies the identity:ϕ(w)1,2 = x′t ⊗ fwt (x) 6= x′t ⊗ fvt (x) = ϕ(v)1,2.

I Setting is = x′s2 , js = x′s−xs

2 ∈ N0 then gives

ϕ(s) = (is − js)⊗(is − js is + js−∞ js − is

I Take ψ(s) = (js − is)⊗ ϕ(s).

The bicyclic monoid

2 ∈ N0 then gives

The bicyclic monoid

2 ∈ N0 then gives

The bicyclic monoid

2 ∈ N0 then gives

The bicyclic monoid

2 ∈ N0 then gives

Variants

I Let T = R,Q or Z and define the semigroup BT := T × Tvia the product

(a, b) · (c, d) = (a− b+ max(b, c), d− c+ max(b, c)).

I For each T as above we have

B ↪→ BT ↪→ UT2(Tmax) ↪→ UT2(T̄max) ↪→ UT2(T).

I Since B and UT2(T) generate the same semigroup variety,it follows that each of the intermediate variants above mustsatisfy exactly the same semigroup identities as these two.

Generalisation

[n] := {1, . . . , n}, Γ ⊆ {(i, j) : i 6= j} a directed graph.

If Γ is transitive, then

Γ(T) = {A ∈Mn(T) : Ai,j 6= −∞⇒ i = j or (i, j) ∈ Γ}

is a semigroup.

Theorem: Let Γ be non-empty, transitive, and acyclic.The semigroup identity w = v holds in Γ(T) if and only iffor all pairs u ∈ Σ+ and ρ a path of length |u| through Γ wehave equality of tropical polynomial functions fwu,ρ = fvu,ρ.

Corollary: Let L(Γ) denote the maximum length of anydirected path in Γ. Then Γ(T) satisfies the same semigroupidentities as UTL(Γ)+1(T).

Example: The free monogenic inverse monoid

I = 〈x, x−1 : xx−1x = x, x−1xx−1 = x−1〉∼= {(i, j, k) ∈ Z3 : i, j > 0,−j ≤ k ≤ i}.

with product

(i, j, k) · (i′, j′, k′) = (max(i, i′ + k), max(j, j′ − k), k + k′)

via x 7→ (1, 0, 1), x−1 7→ (0, 1,−1),

I For Γ = {(1, 3), (2, 3)} can show that I ↪→ Γ(T)

(i, j, k) 7→

k −∞ i−∞ −k j−∞ −∞ 0

I Γ(T) satisfies the same semigroup identities as UT2(T), andhence B.

I Follows that I satisfies all identities satisfied by B.

I Does Mn(T) satisfy a non-trivial semigroup identity?

Izhakian and Margolis: Identity of length 20 for M2(T).Shitov: Identity of length 1,795,308 for M3(T).Open for n > 3.

I Does M2(T) satisfy a shorter semigroup identity?

Daviaud, J, 2017: Yes!(Although not much shorter - minimal length is 17.)

I Can we “describe” the identities satisfied by M2(T)?

Daviaud and J, 2017: Necessary conditions for n = 2; thesemay not yet be sufficient. (Work in progress.)

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