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Introduction MZ1 MZ2 Putting pieces together Absorption and reflexive digraphs Alexandr Kazda, Libor Barto Department of Algebra Charles University, Prague June 2012 Alexandr Kazda, Libor Barto Absorption and reflexive digraphs

SLIDE 1

Introduction MZ1 MZ2 Putting pieces together

Absorption and reflexive digraphs

Alexandr Kazda, Libor Barto

Department of Algebra Charles University, Prague

June 2012

Alexandr Kazda, Libor Barto Absorption and reflexive digraphs

SLIDE 2

Introduction MZ1 MZ2 Putting pieces together

Our goal

M. Mar´
ti and L. Z´

adori: CM ⇒ NU for reflexive digraphs. We show an alternative proof using absorption. All our graphs will be reflexive.

Alexandr Kazda, Libor Barto Absorption and reflexive digraphs

SLIDE 3

Introduction MZ1 MZ2 Putting pieces together

Our goal

M. Mar´
ti and L. Z´

adori: CM ⇒ NU for reflexive digraphs. We show an alternative proof using absorption. All our graphs will be reflexive.

Alexandr Kazda, Libor Barto Absorption and reflexive digraphs

SLIDE 4

Introduction MZ1 MZ2 Putting pieces together

Our goal

M. Mar´
ti and L. Z´

adori: CM ⇒ NU for reflexive digraphs. We show an alternative proof using absorption. All our graphs will be reflexive.

Alexandr Kazda, Libor Barto Absorption and reflexive digraphs

SLIDE 5

Introduction MZ1 MZ2 Putting pieces together

Our goal

M. Mar´
ti and L. Z´

adori: CM ⇒ NU for reflexive digraphs. We show an alternative proof using absorption. All our graphs will be reflexive.

Alexandr Kazda, Libor Barto Absorption and reflexive digraphs

SLIDE 6

Introduction MZ1 MZ2 Putting pieces together

CM ⇒ MZ1 + 2

Let G be a CM reflexive digraph. Then for any K reflexive digraph: MZ1 If H is a connected component of G, R ≤ G K and R ⊂ HK then R is connected. MZ2 If H is a strongly connected component of G, R ≤ G K and R ⊂ HK then R is extremely connected. Mar´

ti and Z´

adori have given a nice proof that MZ1 + 2 implies NU.

Alexandr Kazda, Libor Barto Absorption and reflexive digraphs

SLIDE 7

Introduction MZ1 MZ2 Putting pieces together

CM ⇒ MZ1 + 2

Let G be a CM reflexive digraph. Then for any K reflexive digraph: MZ1 If H is a connected component of G, R ≤ G K and R ⊂ HK then R is connected. MZ2 If H is a strongly connected component of G, R ≤ G K and R ⊂ HK then R is extremely connected. Mar´

ti and Z´

adori have given a nice proof that MZ1 + 2 implies NU.

Alexandr Kazda, Libor Barto Absorption and reflexive digraphs

SLIDE 8

Introduction MZ1 MZ2 Putting pieces together

CM ⇒ MZ1 + 2

Let G be a CM reflexive digraph. Then for any K reflexive digraph: MZ1 If H is a connected component of G, R ≤ G K and R ⊂ HK then R is connected. MZ2 If H is a strongly connected component of G, R ≤ G K and R ⊂ HK then R is extremely connected. Mar´

ti and Z´

adori have given a nice proof that MZ1 + 2 implies NU.

Alexandr Kazda, Libor Barto Absorption and reflexive digraphs

SLIDE 9

Introduction MZ1 MZ2 Putting pieces together

CM ⇒ MZ1 + 2

Let G be a CM reflexive digraph. Then for any K reflexive digraph: MZ1 If H is a connected component of G, R ≤ G K and R ⊂ HK then R is connected. MZ2 If H is a strongly connected component of G, R ≤ G K and R ⊂ HK then R is extremely connected. Mar´

ti and Z´

adori have given a nice proof that MZ1 + 2 implies NU.

Alexandr Kazda, Libor Barto Absorption and reflexive digraphs

SLIDE 10

Introduction MZ1 MZ2 Putting pieces together

G K

The digraph G K has as vertices all the homomorphisms K → G. We have f → g if whenever u → v in K then f (u) → g(v) in G. In particular G K is itself a reflexive digraph. . . . . . that contains a copy of G on the “diagonal”. . . . . . and if G was CM then so is G K.

Alexandr Kazda, Libor Barto Absorption and reflexive digraphs

SLIDE 11

Introduction MZ1 MZ2 Putting pieces together

G K

The digraph G K has as vertices all the homomorphisms K → G. We have f → g if whenever u → v in K then f (u) → g(v) in G. In particular G K is itself a reflexive digraph. . . . . . that contains a copy of G on the “diagonal”. . . . . . and if G was CM then so is G K.

Alexandr Kazda, Libor Barto Absorption and reflexive digraphs

SLIDE 12

Introduction MZ1 MZ2 Putting pieces together

G K

The digraph G K has as vertices all the homomorphisms K → G. We have f → g if whenever u → v in K then f (u) → g(v) in G. In particular G K is itself a reflexive digraph. . . . . . that contains a copy of G on the “diagonal”. . . . . . and if G was CM then so is G K.

Alexandr Kazda, Libor Barto Absorption and reflexive digraphs

SLIDE 13

Introduction MZ1 MZ2 Putting pieces together

G K

The digraph G K has as vertices all the homomorphisms K → G. We have f → g if whenever u → v in K then f (u) → g(v) in G. In particular G K is itself a reflexive digraph. . . . . . that contains a copy of G on the “diagonal”. . . . . . and if G was CM then so is G K.

Alexandr Kazda, Libor Barto Absorption and reflexive digraphs

SLIDE 14

Introduction MZ1 MZ2 Putting pieces together

G K

The digraph G K has as vertices all the homomorphisms K → G. We have f → g if whenever u → v in K then f (u) → g(v) in G. In particular G K is itself a reflexive digraph. . . . . . that contains a copy of G on the “diagonal”. . . . . . and if G was CM then so is G K.

Alexandr Kazda, Libor Barto Absorption and reflexive digraphs

SLIDE 15

Introduction MZ1 MZ2 Putting pieces together

G K

The digraph G K has as vertices all the homomorphisms K → G. We have f → g if whenever u → v in K then f (u) → g(v) in G. In particular G K is itself a reflexive digraph. . . . . . that contains a copy of G on the “diagonal”. . . . . . and if G was CM then so is G K.

Alexandr Kazda, Libor Barto Absorption and reflexive digraphs

SLIDE 16

Introduction MZ1 MZ2 Putting pieces together

Absorption

Let (V , E) be reflexive, U ⊂ V . Assume we have Gumm terms and U g V . Then: If (V , E) is connected then so is (U, E). If (V , E) is strongly connected then so is (U, E). Note: Mar´

ti and Z´

adori actually prove both claims in their paper (without mentioning absorption).

Alexandr Kazda, Libor Barto Absorption and reflexive digraphs

SLIDE 17

Introduction MZ1 MZ2 Putting pieces together

Absorption

Let (V , E) be reflexive, U ⊂ V . Assume we have Gumm terms and U g V . Then: If (V , E) is connected then so is (U, E). If (V , E) is strongly connected then so is (U, E). Note: Mar´

ti and Z´

adori actually prove both claims in their paper (without mentioning absorption).

Alexandr Kazda, Libor Barto Absorption and reflexive digraphs

SLIDE 18

Introduction MZ1 MZ2 Putting pieces together

Absorption

Let (V , E) be reflexive, U ⊂ V . Assume we have Gumm terms and U g V . Then: If (V , E) is connected then so is (U, E). If (V , E) is strongly connected then so is (U, E). Note: Mar´

ti and Z´

adori actually prove both claims in their paper (without mentioning absorption).

Alexandr Kazda, Libor Barto Absorption and reflexive digraphs

SLIDE 19

Introduction MZ1 MZ2 Putting pieces together

Absorption

Let (V , E) be reflexive, U ⊂ V . Assume we have Gumm terms and U g V . Then: If (V , E) is connected then so is (U, E). If (V , E) is strongly connected then so is (U, E). Note: Mar´

ti and Z´

adori actually prove both claims in their paper (without mentioning absorption).

Alexandr Kazda, Libor Barto Absorption and reflexive digraphs

SLIDE 20

Introduction MZ1 MZ2 Putting pieces together

Proving MZ1

Goal: If H is a connected component of G, R ≤ G K and R ⊂ HK then R is connected. We show by induction that R is connected if it contains the diagonal. In the general case, we have some pp definition D of R. If we remove all constant constraints in D we get a pp definition of some S ⊃ R. Now S contains the diagonal and R g S. Therefore, R must be connected.

Alexandr Kazda, Libor Barto Absorption and reflexive digraphs

SLIDE 21

Introduction MZ1 MZ2 Putting pieces together

Proving MZ1

Goal: If H is a connected component of G, R ≤ G K and R ⊂ HK then R is connected. We show by induction that R is connected if it contains the diagonal. In the general case, we have some pp definition D of R. If we remove all constant constraints in D we get a pp definition of some S ⊃ R. Now S contains the diagonal and R g S. Therefore, R must be connected.

Alexandr Kazda, Libor Barto Absorption and reflexive digraphs

SLIDE 22

Introduction MZ1 MZ2 Putting pieces together

Proving MZ1

Goal: If H is a connected component of G, R ≤ G K and R ⊂ HK then R is connected. We show by induction that R is connected if it contains the diagonal. In the general case, we have some pp definition D of R. If we remove all constant constraints in D we get a pp definition of some S ⊃ R. Now S contains the diagonal and R g S. Therefore, R must be connected.

Alexandr Kazda, Libor Barto Absorption and reflexive digraphs

SLIDE 23

Introduction MZ1 MZ2 Putting pieces together

Proving MZ1

Goal: If H is a connected component of G, R ≤ G K and R ⊂ HK then R is connected. We show by induction that R is connected if it contains the diagonal. In the general case, we have some pp definition D of R. If we remove all constant constraints in D we get a pp definition of some S ⊃ R. Now S contains the diagonal and R g S. Therefore, R must be connected.

Alexandr Kazda, Libor Barto Absorption and reflexive digraphs

SLIDE 24

Introduction MZ1 MZ2 Putting pieces together

Proving MZ1

Goal: If H is a connected component of G, R ≤ G K and R ⊂ HK then R is connected. We show by induction that R is connected if it contains the diagonal. In the general case, we have some pp definition D of R. If we remove all constant constraints in D we get a pp definition of some S ⊃ R. Now S contains the diagonal and R g S. Therefore, R must be connected.

Alexandr Kazda, Libor Barto Absorption and reflexive digraphs

SLIDE 25

Introduction MZ1 MZ2 Putting pieces together

MZ1.5

The previous argument can be easily modified to prove that if H is strongly connected then R ⊂ HK is strongly connected. Corollary: Any subalgebra of a strongly connected CM digraph is strongly connected.

Alexandr Kazda, Libor Barto Absorption and reflexive digraphs

SLIDE 26

Introduction MZ1 MZ2 Putting pieces together

MZ1.5

The previous argument can be easily modified to prove that if H is strongly connected then R ⊂ HK is strongly connected. Corollary: Any subalgebra of a strongly connected CM digraph is strongly connected.

Alexandr Kazda, Libor Barto Absorption and reflexive digraphs

SLIDE 27

Introduction MZ1 MZ2 Putting pieces together

MZ1.5

The previous argument can be easily modified to prove that if H is strongly connected then R ⊂ HK is strongly connected. Corollary: Any subalgebra of a strongly connected CM digraph is strongly connected.

Alexandr Kazda, Libor Barto Absorption and reflexive digraphs

SLIDE 28

Introduction MZ1 MZ2 Putting pieces together

Proving MZ2

Goal: If H is a strongly connected component of G, R ≤ G K and R ⊂ HK then R is extremely connected. By the previous argument we know that R is strongly connected. All we need is CM + strongly connected ⇒ all subalgebras extremely connected.

Alexandr Kazda, Libor Barto Absorption and reflexive digraphs

SLIDE 29

Introduction MZ1 MZ2 Putting pieces together

Proving MZ2

Goal: If H is a strongly connected component of G, R ≤ G K and R ⊂ HK then R is extremely connected. By the previous argument we know that R is strongly connected. All we need is CM + strongly connected ⇒ all subalgebras extremely connected.

Alexandr Kazda, Libor Barto Absorption and reflexive digraphs

SLIDE 30

Introduction MZ1 MZ2 Putting pieces together

Proving MZ2

Goal: If H is a strongly connected component of G, R ≤ G K and R ⊂ HK then R is extremely connected. By the previous argument we know that R is strongly connected. All we need is CM + strongly connected ⇒ all subalgebras extremely connected.

Alexandr Kazda, Libor Barto Absorption and reflexive digraphs

SLIDE 31

Introduction MZ1 MZ2 Putting pieces together

Strongly connected ⇒ extremely connected

Take the smallest counterexample G: CM, strongly connected, some subalgebra not extremely connected. MZ1.5: Any subalgebra of G must be strongly connected. By minimality, any proper subalgebra of G must be extremely connected and G is not extremely connected. Singletons are subalgebras ⇒ G is extremely connected.

Alexandr Kazda, Libor Barto Absorption and reflexive digraphs

SLIDE 32

Introduction MZ1 MZ2 Putting pieces together

Strongly connected ⇒ extremely connected

Take the smallest counterexample G: CM, strongly connected, some subalgebra not extremely connected. MZ1.5: Any subalgebra of G must be strongly connected. By minimality, any proper subalgebra of G must be extremely connected and G is not extremely connected. Singletons are subalgebras ⇒ G is extremely connected.

Alexandr Kazda, Libor Barto Absorption and reflexive digraphs

SLIDE 33

Introduction MZ1 MZ2 Putting pieces together

Strongly connected ⇒ extremely connected

Take the smallest counterexample G: CM, strongly connected, some subalgebra not extremely connected. MZ1.5: Any subalgebra of G must be strongly connected. By minimality, any proper subalgebra of G must be extremely connected and G is not extremely connected. Singletons are subalgebras ⇒ G is extremely connected.

Alexandr Kazda, Libor Barto Absorption and reflexive digraphs

SLIDE 34

Introduction MZ1 MZ2 Putting pieces together

Strongly connected ⇒ extremely connected

Take the smallest counterexample G: CM, strongly connected, some subalgebra not extremely connected. MZ1.5: Any subalgebra of G must be strongly connected. By minimality, any proper subalgebra of G must be extremely connected and G is not extremely connected. Singletons are subalgebras ⇒ G is extremely connected.

Alexandr Kazda, Libor Barto Absorption and reflexive digraphs

SLIDE 35

Introduction MZ1 MZ2 Putting pieces together

Strongly connected ⇒ extremely connected

Take the smallest counterexample G: CM, strongly connected, some subalgebra not extremely connected. MZ1.5: Any subalgebra of G must be strongly connected. By minimality, any proper subalgebra of G must be extremely connected and G is not extremely connected. Singletons are subalgebras ⇒ G is extremely connected.

Alexandr Kazda, Libor Barto Absorption and reflexive digraphs

SLIDE 36

Introduction MZ1 MZ2 Putting pieces together

We now have: MZ1 (“Subpowers of connected are connected”) MZ2 (“Subpowers of strongly connected are extremely connected”) MZ1 + 2 implies NU NU

Alexandr Kazda, Libor Barto Absorption and reflexive digraphs

SLIDE 37

Introduction MZ1 MZ2 Putting pieces together

We now have: MZ1 (“Subpowers of connected are connected”) MZ2 (“Subpowers of strongly connected are extremely connected”) MZ1 + 2 implies NU NU

Alexandr Kazda, Libor Barto Absorption and reflexive digraphs

SLIDE 38

Introduction MZ1 MZ2 Putting pieces together

We now have: MZ1 (“Subpowers of connected are connected”) MZ2 (“Subpowers of strongly connected are extremely connected”) MZ1 + 2 implies NU NU

Alexandr Kazda, Libor Barto Absorption and reflexive digraphs

SLIDE 39

Introduction MZ1 MZ2 Putting pieces together

We now have: MZ1 (“Subpowers of connected are connected”) MZ2 (“Subpowers of strongly connected are extremely connected”) MZ1 + 2 implies NU NU

Alexandr Kazda, Libor Barto Absorption and reflexive digraphs

SLIDE 40

Introduction MZ1 MZ2 Putting pieces together

We now have: MZ1 (“Subpowers of connected are connected”) MZ2 (“Subpowers of strongly connected are extremely connected”) MZ1 + 2 implies NU NU

Alexandr Kazda, Libor Barto Absorption and reflexive digraphs

SLIDE 41

Introduction MZ1 MZ2 Putting pieces together

Thanks for your attention.

Alexandr Kazda, Libor Barto Absorption and reflexive digraphs