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Linear sandwich semigroups

Igor Dolinka sc:ala seminar, 10 December 2015

The bad news. . .

. . . all groups will be monoids with no identity.

Igor Dolinka Linear sandwich semigroups 1

The good news. . .

. . . all sandwiches are vegan, halal, kosher, nut free, gluten free. . .

Igor Dolinka Linear sandwich semigroups 2

Joint work of: yours truly. . .

Igor Dolinka Linear sandwich semigroups 3

. . . and James East (FBI)

Igor Dolinka Linear sandwich semigroups 4

. . . errr, I mean, James East (Western Sydney University)

Igor Dolinka Linear sandwich semigroups 5

Sandwiches?

Igor Dolinka Linear sandwich semigroups 6

Sandwiches?

Linear sandwich semigroups (Lyapin, 1960; cf Brown 1955)

Igor Dolinka Linear sandwich semigroups 6

Sandwiches?

Linear sandwich semigroups (Lyapin, 1960; cf Brown 1955)

◮ Let Mmn be the set of all m × n matrices over a field F.

Igor Dolinka Linear sandwich semigroups 6

Sandwiches?

Linear sandwich semigroups (Lyapin, 1960; cf Brown 1955)

◮ Let Mmn be the set of all m × n matrices over a field F. ◮ Fix A ∈ Mnm.

Igor Dolinka Linear sandwich semigroups 6

Sandwiches?

Linear sandwich semigroups (Lyapin, 1960; cf Brown 1955)

◮ Let Mmn be the set of all m × n matrices over a field F. ◮ Fix A ∈ Mnm. ◮ For X, Y ∈ Mmn, define X ⋆ Y = XAY .

Igor Dolinka Linear sandwich semigroups 6

Sandwiches?

Linear sandwich semigroups (Lyapin, 1960; cf Brown 1955)

◮ Let Mmn be the set of all m × n matrices over a field F. ◮ Fix A ∈ Mnm. ◮ For X, Y ∈ Mmn, define X ⋆ Y = XAY . ◮ MA mn = (Mmn, ⋆) is a linear sandwich semigroup.

Igor Dolinka Linear sandwich semigroups 6

Sandwiches?

Linear sandwich semigroups (Lyapin, 1960; cf Brown 1955)

◮ Let Mmn be the set of all m × n matrices over a field F. ◮ Fix A ∈ Mnm. ◮ For X, Y ∈ Mmn, define X ⋆ Y = XAY . ◮ MA mn = (Mmn, ⋆) is a linear sandwich semigroup.

Example

If m = n and A = I, then MA

mn = Mn is the full linear monoid.

Igor Dolinka Linear sandwich semigroups 6

Plan (non-linear)

Igor Dolinka Linear sandwich semigroups 7

Plan (non-linear)

◮ Green’s relations

Igor Dolinka Linear sandwich semigroups 7

Plan (non-linear)

◮ Green’s relations ◮ regular elements

Igor Dolinka Linear sandwich semigroups 7

Plan (non-linear)

◮ Green’s relations ◮ regular elements ◮ ideals

Igor Dolinka Linear sandwich semigroups 7

Plan (non-linear)

◮ Green’s relations ◮ regular elements ◮ ideals ◮ idempotent generation

Igor Dolinka Linear sandwich semigroups 7

Plan (non-linear)

◮ Green’s relations ◮ regular elements ◮ ideals ◮ idempotent generation ◮ small (idempotent) generating sets

Igor Dolinka Linear sandwich semigroups 7

Plan (non-linear)

◮ Green’s relations ◮ regular elements ◮ ideals ◮ idempotent generation ◮ small (idempotent) generating sets ◮ bigger sandwiches?

Igor Dolinka Linear sandwich semigroups 7

Egg-box diagrams for Mn (F = Z2)

1 1 1 1 1 1 1 2

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3

Igor Dolinka Linear sandwich semigroups 8

Egg-box diagrams for Mn (F = Z2)

1 1 1 1 1 1 1 2

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3

Within a   

box row column

  , matrices have same   

rank column space row space

  .

Igor Dolinka Linear sandwich semigroups 8

Egg-box diagrams for Mn (F = Z2)

1 1 1 1 1 1 1 2

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3

Within a   

box row column

  , matrices are   

D-related R-related L -related

  .

Igor Dolinka Linear sandwich semigroups 8

Egg-box diagrams for Mn (F = Z2)

1 1 1 1 1 1 1 2

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3

◮ Cells with an idempotent matrix are shaded. ◮ These are subgroups of Mn(F) isomorphic to GL(r, F).

Igor Dolinka Linear sandwich semigroups 8

Egg sandwiches — MA

mn

1 1 1 1 1 1 1 1

Igor Dolinka Linear sandwich semigroups 9

Egg sandwiches — MA

mn

1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

Igor Dolinka Linear sandwich semigroups 10

Easy lemmas

Lemma

If A, B ∈ Mnm and rank(A) = rank(B), then MA

mn ∼

= MB

mn.

Igor Dolinka Linear sandwich semigroups 11

Easy lemmas

Lemma

If A, B ∈ Mnm and rank(A) = rank(B), then MA

mn ∼

= MB

mn. ◮ So we study MA mn, where

J = Jr = Ir O

O O

• ∈ Mnm

(0 ≤ r ≤ min(m, n) is fixed).

Igor Dolinka Linear sandwich semigroups 11

Easy lemmas

Lemma

If A, B ∈ Mnm and rank(A) = rank(B), then MA

mn ∼

= MB

mn. ◮ So we study MA mn, where

J = Jr = Ir O

O O

• ∈ Mnm

(0 ≤ r ≤ min(m, n) is fixed).

◮ Write elements of Mmn in 2 × 2 block form:

A B

C D

• ,

where A ∈ Mrr, B ∈ Mr,n−r, etc.

Igor Dolinka Linear sandwich semigroups 11

Easy lemmas

Lemma

If A, B ∈ Mnm and rank(A) = rank(B), then MA

mn ∼

= MB

mn. ◮ So we study MA mn, where

J = Jr = Ir O

O O

• ∈ Mnm

(0 ≤ r ≤ min(m, n) is fixed).

◮ Write elements of Mmn in 2 × 2 block form:

A B

C D

• ,

where A ∈ Mrr, B ∈ Mr,n−r, etc.

Lemma

A B

C D

• ⋆

E F

G H

• =

AE AF

CE CF

• .

Igor Dolinka Linear sandwich semigroups 11

Green’s relations

Igor Dolinka Linear sandwich semigroups 12

Green’s relations

Let X, Y ∈ Mmn. Write

◮ XRY ⇔ XMn = Y Mn,

Igor Dolinka Linear sandwich semigroups 12

Green’s relations

Let X, Y ∈ Mmn. Write

◮ XRY ⇔ XMn = Y Mn, ◮ XL Y ⇔ MmX = MmY ,

Igor Dolinka Linear sandwich semigroups 12

Green’s relations

Let X, Y ∈ Mmn. Write

◮ XRY ⇔ XMn = Y Mn, ◮ XL Y ⇔ MmX = MmY , ◮ XJ Y ⇔ MmXMn = MmY Mn,

Igor Dolinka Linear sandwich semigroups 12

Green’s relations

Let X, Y ∈ Mmn. Write

◮ XRY ⇔ XMn = Y Mn, ◮ XL Y ⇔ MmX = MmY , ◮ XJ Y ⇔ MmXMn = MmY Mn, ◮ H = R ∩ L , ◮ D = R ∨ L .

Igor Dolinka Linear sandwich semigroups 12

Green’s relations

Let X, Y ∈ Mmn. Write

◮ XRY ⇔ XMn = Y Mn, ◮ XL Y ⇔ MmX = MmY , ◮ XJ Y ⇔ MmXMn = MmY Mn, ◮ H = R ∩ L , ◮ D = R ∨ L .

For X ∈ Mmn, write RX = {Y ∈ Mmn : XRY }, etc.

Igor Dolinka Linear sandwich semigroups 12

Green’s relations

Let X, Y ∈ Mmn. Write

◮ XRY ⇔ XMn = Y Mn, ◮ XL Y ⇔ MmX = MmY , ◮ XJ Y ⇔ MmXMn = MmY Mn, ◮ H = R ∩ L , ◮ D = R ∨ L .

For X ∈ Mmn, write RX = {Y ∈ Mmn : XRY }, etc.

Proposition

◮ RX = {Y ∈ Mmn : Col(X) = Col(Y )}, ◮ LX = {Y ∈ Mmn : Row(X) = Row(Y )}, ◮ JX = DX = {Y ∈ Mmn : rank(X) = rank(Y )}

Igor Dolinka Linear sandwich semigroups 12

Green’s relations

Let X, Y ∈ Mmn. Write

◮ XRY ⇔ X ⋆ Mmn = Y ⋆ Mmn, — i.e., XJMmn = YJMmn,

Igor Dolinka Linear sandwich semigroups 13

Green’s relations

Let X, Y ∈ Mmn. Write

◮ XRY ⇔ X ⋆ Mmn = Y ⋆ Mmn, — i.e., XJMmn = YJMmn, ◮ XL Y ⇔ Mmn ⋆ X = Mmn ⋆ Y ,

Igor Dolinka Linear sandwich semigroups 13

Green’s relations

Let X, Y ∈ Mmn. Write

◮ XRY ⇔ X ⋆ Mmn = Y ⋆ Mmn, — i.e., XJMmn = YJMmn, ◮ XL Y ⇔ Mmn ⋆ X = Mmn ⋆ Y , ◮ XJ Y ⇔ Mmn ⋆ X ⋆ Mmn = Mmn ⋆ Y ⋆ Mmn,

Igor Dolinka Linear sandwich semigroups 13

Green’s relations

Let X, Y ∈ Mmn. Write

◮ XRY ⇔ X ⋆ Mmn = Y ⋆ Mmn, — i.e., XJMmn = YJMmn, ◮ XL Y ⇔ Mmn ⋆ X = Mmn ⋆ Y , ◮ XJ Y ⇔ Mmn ⋆ X ⋆ Mmn = Mmn ⋆ Y ⋆ Mmn, ◮ H = R ∩ L , ◮ D = R ∨ L .

Igor Dolinka Linear sandwich semigroups 13

Green’s relations

Let X, Y ∈ Mmn. Write

◮ XRY ⇔ X ⋆ Mmn = Y ⋆ Mmn, — i.e., XJMmn = YJMmn, ◮ XL Y ⇔ Mmn ⋆ X = Mmn ⋆ Y , ◮ XJ Y ⇔ Mmn ⋆ X ⋆ Mmn = Mmn ⋆ Y ⋆ Mmn, ◮ H = R ∩ L , ◮ D = R ∨ L .

These are the usual Green’s relations on MA

mn.

Igor Dolinka Linear sandwich semigroups 13

Green’s relations

Let X, Y ∈ Mmn. Write

◮ XRY ⇔ X ⋆ Mmn = Y ⋆ Mmn, — i.e., XJMmn = YJMmn, ◮ XL Y ⇔ Mmn ⋆ X = Mmn ⋆ Y , ◮ XJ Y ⇔ Mmn ⋆ X ⋆ Mmn = Mmn ⋆ Y ⋆ Mmn, ◮ H = R ∩ L , ◮ D = R ∨ L .

These are the usual Green’s relations on MA

mn.

For X ∈ Mmn, write RJ

X = {Y ∈ Mmn : XRY }, etc.

Igor Dolinka Linear sandwich semigroups 13

Green’s relations

Let X = A B

C D

• ∈ Mmn.

Igor Dolinka Linear sandwich semigroups 14

Green’s relations

Let X = A B

C D

• ∈ Mmn.

Easy to check:

◮ XJ =

A O

C O

• ,

JX = A B

O O

• ,

JXJ = A O

O O

• .

Igor Dolinka Linear sandwich semigroups 14

Green’s relations

Let X = A B

C D

• ∈ Mmn.

Easy to check:

◮ XJ =

A O

C O

• ,

JX = A B

O O

• ,

JXJ = A O

O O

• .

Define sets

◮ P1 = {X ∈ Mmn : Col(XJ) = Col(X)},

Igor Dolinka Linear sandwich semigroups 14

Green’s relations

Let X = A B

C D

• ∈ Mmn.

Easy to check:

◮ XJ =

A O

C O

• ,

JX = A B

O O

• ,

JXJ = A O

O O

• .

Define sets

◮ P1 = {X ∈ Mmn : Col(XJ) = Col(X)}, ◮ P2 = {X ∈ Mmn : Row(JX) = Row(X)},

Igor Dolinka Linear sandwich semigroups 14

Green’s relations

Let X = A B

C D

• ∈ Mmn.

Easy to check:

◮ XJ =

A O

C O

• ,

JX = A B

O O

• ,

JXJ = A O

O O

• .

Define sets

◮ P1 = {X ∈ Mmn : Col(XJ) = Col(X)}, ◮ P2 = {X ∈ Mmn : Row(JX) = Row(X)}, ◮ P = P1 ∩ P2

Igor Dolinka Linear sandwich semigroups 14

Green’s relations

Let X = A B

C D

• ∈ Mmn.

Easy to check:

◮ XJ =

A O

C O

• ,

JX = A B

O O

• ,

JXJ = A O

O O

• .

Define sets

◮ P1 = {X ∈ Mmn : Col(XJ) = Col(X)}, ◮ P2 = {X ∈ Mmn : Row(JX) = Row(X)}, ◮ P = P1 ∩ P2 = {X ∈ Mmn : rank(JXJ) = rank(X)}

Igor Dolinka Linear sandwich semigroups 14

Green’s relations

Let X = A B

C D

• ∈ Mmn.

Easy to check:

◮ XJ =

A O

C O

• ,

JX = A B

O O

• ,

JXJ = A O

O O

• .

Define sets

◮ P1 = {X ∈ Mmn : Col(XJ) = Col(X)}, ◮ P2 = {X ∈ Mmn : Row(JX) = Row(X)}, ◮ P = P1 ∩ P2 = {X ∈ Mmn : rank(JXJ) = rank(X)}

= Reg(MA

mn)

Igor Dolinka Linear sandwich semigroups 14

Green’s relations

Let X = A B

C D

• ∈ Mmn.

Easy to check:

◮ XJ =

A O

C O

• ,

JX = A B

O O

• ,

JXJ = A O

O O

• .

Define sets

◮ P1 = {X ∈ Mmn : Col(XJ) = Col(X)}, ◮ P2 = {X ∈ Mmn : Row(JX) = Row(X)}, ◮ P = P1 ∩ P2 = {X ∈ Mmn : rank(JXJ) = rank(X)}

= Reg(MA

mn) ≤ MA mn.

Igor Dolinka Linear sandwich semigroups 14

Green’s relations

Proposition

For X ∈ Mmn,

◮ RJ X =

• RX ∩ P1

if X ∈ P1 {X} if X ∈ Mmn \ P1,

Igor Dolinka Linear sandwich semigroups 15

Green’s relations

Proposition

For X ∈ Mmn,

◮ RJ X =

• RX ∩ P1

if X ∈ P1 {X} if X ∈ Mmn \ P1,

◮ LJ X =

• LX ∩ P2

if X ∈ P2 {X} if X ∈ Mmn \ P2,

Igor Dolinka Linear sandwich semigroups 15

Green’s relations

Proposition

For X ∈ Mmn,

◮ RJ X =

• RX ∩ P1

if X ∈ P1 {X} if X ∈ Mmn \ P1,

◮ LJ X =

• LX ∩ P2

if X ∈ P2 {X} if X ∈ Mmn \ P2,

◮ HJ X =

• HX

if X ∈ P {X} if X ∈ Mmn \ P,

Igor Dolinka Linear sandwich semigroups 15

Green’s relations

Proposition (continued)

For X ∈ Mmn,

◮ DJ X =

           DX ∩ P if X ∈ P LX ∩ P2 if X ∈ P2 \ P1 RX ∩ P1 if X ∈ P1 \ P2 {X} if X ∈ Mmn \ (P1 ∪ P2).

Igor Dolinka Linear sandwich semigroups 16

High energy semigroup theory — from Mmn to MA

mn

P1 ¬P1 P2 ¬P2

Igor Dolinka Linear sandwich semigroups 17

High energy semigroup theory — from Mmn to MA

mn

P1 ¬P1 P2 ¬P2

Igor Dolinka Linear sandwich semigroups 17

Small generating sets

Igor Dolinka Linear sandwich semigroups 18

Small generating sets

Theorem

Suppose r = rank(A) < min(m, n).

1 1 1 1 1 1 1 1

Igor Dolinka Linear sandwich semigroups 18

Small generating sets

Theorem

Suppose r = rank(A) < min(m, n).

◮ The D-maximal elements generate MA mn.

1 1 1 1 1 1 1 1

Igor Dolinka Linear sandwich semigroups 18

Small generating sets

Theorem

Suppose r = rank(A) < min(m, n).

◮ The D-maximal elements generate MA mn. ◮ Any generating set must contain all D-maximal elements.

1 1 1 1 1 1 1 1

Igor Dolinka Linear sandwich semigroups 18

Small generating sets

Theorem

Suppose r = rank(A) < min(m, n).

◮ The D-maximal elements generate MA mn. ◮ Any generating set must contain all D-maximal elements. ◮ rank(MA mn) = min(m,n)

• s=r+1

[ m

s ]q [ n s ]q q(s

2)(q − 1)s[s]q!.

1 1 1 1 1 1 1 1

Igor Dolinka Linear sandwich semigroups 18

Small generating sets

Theorem

Suppose r = rank(A) = min(m, n).

1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

Igor Dolinka Linear sandwich semigroups 19

Small generating sets

Theorem

Suppose r = rank(A) = min(m, n).

◮ MA mn has a unique maximal D-class — a rectangular group.

1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

Igor Dolinka Linear sandwich semigroups 19

Small generating sets

Theorem

Suppose r = rank(A) = min(m, n).

◮ MA mn has a unique maximal D-class — a rectangular group. ◮ rank(MA mn) =

• max(m,n)

min(m,n)

• q

.

1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

Igor Dolinka Linear sandwich semigroups 19

Unscrambling the egg — regular elements of MA1

32

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

Igor Dolinka Linear sandwich semigroups 20

Unscrambling the egg — regular elements of MA2

33

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

Igor Dolinka Linear sandwich semigroups 21

Look familiar?

1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

1 1 1 1 1 1 2

Reg(MA1

32)

M1 Reg(MA2

33)

M2

Igor Dolinka Linear sandwich semigroups 22

Look familiar?

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

1 1 1 1 1 1 2

Reg(MA2

33)

Reg(MA2

34)

Reg(MA2

43)

M2

Igor Dolinka Linear sandwich semigroups 23

Look familiar?

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3

Reg(MA3

43)

Reg(MA3

34)

M3

Igor Dolinka Linear sandwich semigroups 24

Big bang — structure of Reg(MA

mn)

Theorem

◮ P = Reg(MA mn) ≤ MA mn is an “inflated Mr” (r = rank(A)).

Igor Dolinka Linear sandwich semigroups 25

Big bang — structure of Reg(MA

mn)

Theorem

◮ P = Reg(MA mn) ≤ MA mn is an “inflated Mr” (r = rank(A)). ◮ Mr has a chain of D-classes: D0 < D1 < · · · < Dr.

Igor Dolinka Linear sandwich semigroups 25

Big bang — structure of Reg(MA

mn)

Theorem

◮ P = Reg(MA mn) ≤ MA mn is an “inflated Mr” (r = rank(A)). ◮ Mr has a chain of D-classes: D0 < D1 < · · · < Dr. ◮ P has a chain of D-classes: D0 < D1 < · · · < Dr.

Igor Dolinka Linear sandwich semigroups 25

Big bang — structure of Reg(MA

mn)

Theorem

◮ P = Reg(MA mn) ≤ MA mn is an “inflated Mr” (r = rank(A)). ◮ Mr has a chain of D-classes: D0 < D1 < · · · < Dr. ◮ P has a chain of D-classes: D0 < D1 < · · · < Dr. ◮ Ds(Mr) maps onto [ r s ]q R-classes of Mr.

Igor Dolinka Linear sandwich semigroups 25

Big bang — structure of Reg(MA

mn)

Theorem

◮ P = Reg(MA mn) ≤ MA mn is an “inflated Mr” (r = rank(A)). ◮ Mr has a chain of D-classes: D0 < D1 < · · · < Dr. ◮ P has a chain of D-classes: D0 < D1 < · · · < Dr. ◮ Ds(Mr) maps onto [ r s ]q R-classes of Mr. ◮ Each of these expands into qs(m−r) R-classes in DA s (P).

Igor Dolinka Linear sandwich semigroups 25

Big bang — structure of Reg(MA

mn)

Theorem

◮ P = Reg(MA mn) ≤ MA mn is an “inflated Mr” (r = rank(A)). ◮ Mr has a chain of D-classes: D0 < D1 < · · · < Dr. ◮ P has a chain of D-classes: D0 < D1 < · · · < Dr. ◮ Ds(Mr) maps onto [ r s ]q R-classes of Mr. ◮ Each of these expands into qs(m−r) R-classes in DA s (P). ◮ Group H -classes in Ds(Mr) and DA s (P) are ∼

= Gs.

Igor Dolinka Linear sandwich semigroups 25

Big bang — structure of Reg(MA

mn)

Theorem

◮ P = Reg(MA mn) ≤ MA mn is an “inflated Mr” (r = rank(A)). ◮ Mr has a chain of D-classes: D0 < D1 < · · · < Dr. ◮ P has a chain of D-classes: D0 < D1 < · · · < Dr. ◮ Ds(Mr) maps onto [ r s ]q R-classes of Mr. ◮ Each of these expands into qs(m−r) R-classes in DA s (P). ◮ Group H -classes in Ds(Mr) and DA s (P) are ∼

= Gs.

◮ |P| = r s=0 qs(m+n−2r)q(s

2)(q − 1)s[s]q! [ r

s ]2 q.

Igor Dolinka Linear sandwich semigroups 25

Big bang — structure of Reg(MA

mn)

Theorem

◮ P = Reg(MA mn) ≤ MA mn is an “inflated Mr” (r = rank(A)). ◮ Mr has a chain of D-classes: D0 < D1 < · · · < Dr. ◮ P has a chain of D-classes: D0 < D1 < · · · < Dr. ◮ Ds(Mr) maps onto [ r s ]q R-classes of Mr. ◮ Each of these expands into qs(m−r) R-classes in DA s (P). ◮ Group H -classes in Ds(Mr) and DA s (P) are ∼

= Gs.

◮ |P| = r s=0 qs(m+n−2r)q(s

2)(q − 1)s[s]q! [ r

s ]2 q. ◮ rank(P) = qr(max(m,n)−r) + 1.

Igor Dolinka Linear sandwich semigroups 25

Idempotent generators

Theorem

◮

E(Mr)

• = r

s=0 qs(r−s) [ r s ]q.

Igor Dolinka Linear sandwich semigroups 26

Idempotent generators

Theorem

◮

E(Mr)

• = r

s=0 qs(r−s) [ r s ]q. ◮

E(MA

mn)

• = r

s=0 qs(m+n−r−s) [ r s ]q.

Igor Dolinka Linear sandwich semigroups 26

Idempotent generators

Theorem

◮

E(Mr)

• = r

s=0 qs(r−s) [ r s ]q. ◮

E(MA

mn)

• = r

s=0 qs(m+n−r−s) [ r s ]q. ◮ Er =

• E(Mr)
• = {Ir} ∪ (Mr \ Gr) — Erdos.

Igor Dolinka Linear sandwich semigroups 26

Idempotent generators

Theorem

◮

E(Mr)

• = r

s=0 qs(r−s) [ r s ]q. ◮

E(MA

mn)

• = r

s=0 qs(m+n−r−s) [ r s ]q. ◮ Er =

• E(Mr)
• = {Ir} ∪ (Mr \ Gr) — Erdos.

◮ rank(Er) = idrank(Er) = 1 + (qr − 1)/(q − 1) — Dawlings.

Igor Dolinka Linear sandwich semigroups 26

Idempotent generators

Theorem

◮

E(Mr)

• = r

s=0 qs(r−s) [ r s ]q. ◮

E(MA

mn)

• = r

s=0 qs(m+n−r−s) [ r s ]q. ◮ Er =

• E(Mr)
• = {Ir} ∪ (Mr \ Gr) — Erdos.

◮ rank(Er) = idrank(Er) = 1 + (qr − 1)/(q − 1) — Dawlings. ◮ EA mn =

• E(MA

mn)

• = E(Dr) ∪ (P \ Dr).

Igor Dolinka Linear sandwich semigroups 26

Idempotent generators

Theorem

◮

E(Mr)

• = r

s=0 qs(r−s) [ r s ]q. ◮

E(MA

mn)

• = r

s=0 qs(m+n−r−s) [ r s ]q. ◮ Er =

• E(Mr)
• = {Ir} ∪ (Mr \ Gr) — Erdos.

◮ rank(Er) = idrank(Er) = 1 + (qr − 1)/(q − 1) — Dawlings. ◮ EA mn =

• E(MA

mn)

• = E(Dr) ∪ (P \ Dr).

◮ rank(EA mn) = idrank(EA mn) = qr(max(m,n)−r) + (qr − 1)/(q − 1).

Igor Dolinka Linear sandwich semigroups 26

Idempotent generators

Theorem

◮

E(Mr)

• = r

s=0 qs(r−s) [ r s ]q. ◮

E(MA

mn)

• = r

s=0 qs(m+n−r−s) [ r s ]q. ◮ Er =

• E(Mr)
• = {Ir} ∪ (Mr \ Gr) — Erdos.

◮ rank(Er) = idrank(Er) = 1 + (qr − 1)/(q − 1) — Dawlings. ◮ EA mn =

• E(MA

mn)

• = E(Dr) ∪ (P \ Dr).

◮ rank(EA mn) = iDrank(EA mn) = qr(max(m,n)−r) + (qr − 1)/(q − 1).

Igor Dolinka Linear sandwich semigroups 26

Ideals

Theorem

◮ The ideals of Mr form a chain:

{Or} = I0 ⊂ I1 ⊂ · · · ⊂ Ir = Mr.

Igor Dolinka Linear sandwich semigroups 27

Ideals

Theorem

◮ The ideals of Mr form a chain:

{Or} = I0 ⊂ I1 ⊂ · · · ⊂ Ir = Mr.

◮ rank(Is) = idrank(Is) = [ r s ]q for 0 ≤ s < r — Gray.

Igor Dolinka Linear sandwich semigroups 27

Ideals

Theorem

◮ The ideals of Mr form a chain:

{Or} = I0 ⊂ I1 ⊂ · · · ⊂ Ir = Mr.

◮ rank(Is) = idrank(Is) = [ r s ]q for 0 ≤ s < r — Gray. ◮ The ideals of P = Reg(MA mn) form a chain:

{O} = I0 ⊂ I1 ⊂ · · · ⊂ Ir = P.

Igor Dolinka Linear sandwich semigroups 27

Ideals

Theorem

◮ The ideals of Mr form a chain:

{Or} = I0 ⊂ I1 ⊂ · · · ⊂ Ir = Mr.

◮ rank(Is) = idrank(Is) = [ r s ]q for 0 ≤ s < r — Gray. ◮ The ideals of P = Reg(MA mn) form a chain:

{O} = I0 ⊂ I1 ⊂ · · · ⊂ Ir = P.

◮ rank(Is) = idrank(Is) = qs(max(m,n)−r) [ r s ]q for 0 ≤ s < r.

Igor Dolinka Linear sandwich semigroups 27

More sandwiches. . .

Igor Dolinka Linear sandwich semigroups 28

More sandwiches. . .

◮ Let C = (Ob, Hom) be a small category.

Igor Dolinka Linear sandwich semigroups 28

More sandwiches. . .

◮ Let C = (Ob, Hom) be a small category. ◮ If θ ∈ Hom(Y , X) is fixed, Hom(X, Y ) is a semigroup under:

f ⋆ g = f ◦ θ ◦ g.

Igor Dolinka Linear sandwich semigroups 28

More sandwiches. . .

◮ Let C = (Ob, Hom) be a small category. ◮ If θ ∈ Hom(Y , X) is fixed, Hom(X, Y ) is a semigroup under:

f ⋆ g = f ◦ θ ◦ g.

◮ MA mn arises when C is a linear category.

Igor Dolinka Linear sandwich semigroups 28

More sandwiches. . .

◮ Let C = (Ob, Hom) be a small category. ◮ If θ ∈ Hom(Y , X) is fixed, Hom(X, Y ) is a semigroup under:

f ⋆ g = f ◦ θ ◦ g.

◮ MA mn arises when C is a linear category. ◮ Work is under way for diagram categories and more. . .

Igor Dolinka Linear sandwich semigroups 28

Thank you!

◮ Variants of finite full transformation semigroups

◮ Dolinka and East — http://arxiv.org/abs/1410.5253

◮ Semigroups of rectangular matrices under a sandwich operation

◮ Dolinka and East — http://arxiv.org/abs/1503.03139 Igor Dolinka Linear sandwich semigroups 29