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Diagram Semigroups Much more fun than transformations! Michael - - PowerPoint PPT Presentation

Apr 27, 2023 193 likes •724 views

Diagram Semigroups Much more fun than transformations! Michael Torpey University of St Andrews 2017-06-07 Michael Torpey (University of St Andrews) Diagram Semigroups 2017-06-07 1 / 8 Transformations Michael Torpey (University of St

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Diagram Semigroups

Much more fun than transformations! Michael Torpey

University of St Andrews

2017-06-07

Michael Torpey (University of St Andrews) Diagram Semigroups 2017-06-07 1 / 8

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Transformations

Michael Torpey (University of St Andrews) Diagram Semigroups 2017-06-07 2 / 8

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Transformations

Definition

A transformation on a set X is any function τ : X → X

Michael Torpey (University of St Andrews) Diagram Semigroups 2017-06-07 2 / 8

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Transformations

Definition

A transformation on a set X is any function τ : X → X Assume X = n = {1, 2, . . . , n}, and write TX as Tn.

Michael Torpey (University of St Andrews) Diagram Semigroups 2017-06-07 2 / 8

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Transformations

Definition

A transformation on a set X is any function τ : X → X Assume X = n = {1, 2, . . . , n}, and write TX as Tn.

Theorem (Cayley for semigroups)

Every semigroup S is isomorphic to a subsemigroup of the full transformation semigroup TS. [1, p.7]

Michael Torpey (University of St Andrews) Diagram Semigroups 2017-06-07 2 / 8

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Transformations

Definition

A transformation on a set X is any function τ : X → X Assume X = n = {1, 2, . . . , n}, and write TX as Tn.

Theorem (Cayley for semigroups)

Every semigroup S is isomorphic to a subsemigroup of the full transformation semigroup TS. [1, p.7] α =

1

2 3 4 5 1 3 1 5 5

,

β =

1

2 3 4 5 3 1 3 3 5

Michael Torpey (University of St Andrews)

Diagram Semigroups 2017-06-07 2 / 8

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Transformations

Definition

A transformation on a set X is any function τ : X → X Assume X = n = {1, 2, . . . , n}, and write TX as Tn.

Theorem (Cayley for semigroups)

Every semigroup S is isomorphic to a subsemigroup of the full transformation semigroup TS. [1, p.7] α =

1

2 3 4 5 1 3 1 5 5

,

β =

1

2 3 4 5 3 1 3 3 5

α =

, β = ,

Michael Torpey (University of St Andrews) Diagram Semigroups 2017-06-07 2 / 8

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Transformations

Definition

A transformation on a set X is any function τ : X → X Assume X = n = {1, 2, . . . , n}, and write TX as Tn.

Theorem (Cayley for semigroups)

Every semigroup S is isomorphic to a subsemigroup of the full transformation semigroup TS. [1, p.7] α =

1

2 3 4 5 1 3 1 5 5

,

β =

1

2 3 4 5 3 1 3 3 5

α =

, β = , αβ =

Michael Torpey (University of St Andrews) Diagram Semigroups 2017-06-07 2 / 8

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Transformations

Definition

A transformation on a set X is any function τ : X → X Assume X = n = {1, 2, . . . , n}, and write TX as Tn.

Theorem (Cayley for semigroups)

Every semigroup S is isomorphic to a subsemigroup of the full transformation semigroup TS. [1, p.7] α =

1

2 3 4 5 1 3 1 5 5

,

β =

1

2 3 4 5 3 1 3 3 5

α =

, β = , αβ = =

Michael Torpey (University of St Andrews) Diagram Semigroups 2017-06-07 2 / 8

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Partitions

Michael Torpey (University of St Andrews) Diagram Semigroups 2017-06-07 3 / 8

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Partitions

Pn

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Partitions

Pn – the (bi)partition monoid

Michael Torpey (University of St Andrews) Diagram Semigroups 2017-06-07 3 / 8

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Partitions

Pn – the (bi)partition monoid α = ,

Michael Torpey (University of St Andrews) Diagram Semigroups 2017-06-07 3 / 8

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Partitions

Pn – the (bi)partition monoid α = ,

Michael Torpey (University of St Andrews) Diagram Semigroups 2017-06-07 3 / 8

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Partitions

Pn – the (bi)partition monoid α = , β =

Michael Torpey (University of St Andrews) Diagram Semigroups 2017-06-07 3 / 8

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Partitions

Pn – the (bi)partition monoid α = , β =

Definition

A (bi)partition is any equivalence relation on n ∪ n′, where n′ = {1′, 2′, . . . , n′}.

Michael Torpey (University of St Andrews) Diagram Semigroups 2017-06-07 3 / 8

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Partitions

Pn – the (bi)partition monoid α = , β =

Definition

A (bi)partition is any equivalence relation on n ∪ n′, where n′ = {1′, 2′, . . . , n′}. γ = ,

Michael Torpey (University of St Andrews) Diagram Semigroups 2017-06-07 3 / 8

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Partitions

Pn – the (bi)partition monoid α = , β =

Definition

A (bi)partition is any equivalence relation on n ∪ n′, where n′ = {1′, 2′, . . . , n′}. γ = , δ =

Michael Torpey (University of St Andrews) Diagram Semigroups 2017-06-07 3 / 8

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Partitions

Pn – the (bi)partition monoid α = , β =

Definition

A (bi)partition is any equivalence relation on n ∪ n′, where n′ = {1′, 2′, . . . , n′}. γ = , δ =

Michael Torpey (University of St Andrews) Diagram Semigroups 2017-06-07 3 / 8

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Partitions

Pn – the (bi)partition monoid α = , β =

Definition

A (bi)partition is any equivalence relation on n ∪ n′, where n′ = {1′, 2′, . . . , n′}. γ = , δ = γδ =

Michael Torpey (University of St Andrews) Diagram Semigroups 2017-06-07 3 / 8

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Partitions

Pn – the (bi)partition monoid α = , β =

Definition

A (bi)partition is any equivalence relation on n ∪ n′, where n′ = {1′, 2′, . . . , n′}. γ = , δ = γδ =

Michael Torpey (University of St Andrews) Diagram Semigroups 2017-06-07 3 / 8

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Partitions

Pn – the (bi)partition monoid α = , β =

Definition

A (bi)partition is any equivalence relation on n ∪ n′, where n′ = {1′, 2′, . . . , n′}. γ = , δ = γδ = =

Michael Torpey (University of St Andrews) Diagram Semigroups 2017-06-07 3 / 8

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Partitions

Pn – the (bi)partition monoid α = , β =

Definition

A (bi)partition is any equivalence relation on n ∪ n′, where n′ = {1′, 2′, . . . , n′}. γ = , δ = γδ = = =

Michael Torpey (University of St Andrews) Diagram Semigroups 2017-06-07 3 / 8

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How big is the partition monoid?

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How big is the partition monoid?

Pn consists of all partitions of 2n points.

Michael Torpey (University of St Andrews) Diagram Semigroups 2017-06-07 4 / 8

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How big is the partition monoid?

Pn consists of all partitions of 2n points. Its size is the Bell number B2n.

Michael Torpey (University of St Andrews) Diagram Semigroups 2017-06-07 4 / 8

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How big is the partition monoid?

Pn consists of all partitions of 2n points. Its size is the Bell number B2n. n 1 2 3 4 5 6 7 . . . |Pn| 2 15 203 4,140 115,975 4,213,597 190,899,322 . . .

Table: Sizes of partition monoids

Michael Torpey (University of St Andrews) Diagram Semigroups 2017-06-07 4 / 8

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How big is the partition monoid?

Pn consists of all partitions of 2n points. Its size is the Bell number B2n. n 1 2 3 4 5 6 7 . . . |Pn| 2 15 203 4,140 115,975 4,213,597 190,899,322 . . .

Table: Sizes of partition monoids

|P10| ≈ 5.2 × 1013.

Michael Torpey (University of St Andrews) Diagram Semigroups 2017-06-07 4 / 8

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Attributes of partitions

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Attributes of partitions

Definition

A block in a partition α ∈ Pn is transversal if it contains points from both n and n′ (i.e. it “crosses the diagram”).

Michael Torpey (University of St Andrews) Diagram Semigroups 2017-06-07 5 / 8

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Attributes of partitions

Definition

A block in a partition α ∈ Pn is transversal if it contains points from both n and n′ (i.e. it “crosses the diagram”).

Definition

The rank of a partition is the number of transversal blocks it has.

Michael Torpey (University of St Andrews) Diagram Semigroups 2017-06-07 5 / 8

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Attributes of partitions

Definition

A block in a partition α ∈ Pn is transversal if it contains points from both n and n′ (i.e. it “crosses the diagram”).

Definition

The rank of a partition is the number of transversal blocks it has. α = β =

Michael Torpey (University of St Andrews) Diagram Semigroups 2017-06-07 5 / 8

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Attributes of partitions

Definition

A block in a partition α ∈ Pn is transversal if it contains points from both n and n′ (i.e. it “crosses the diagram”).

Definition

The rank of a partition is the number of transversal blocks it has. α = β = rank(α) = 1,

Michael Torpey (University of St Andrews) Diagram Semigroups 2017-06-07 5 / 8

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Attributes of partitions

Definition

A block in a partition α ∈ Pn is transversal if it contains points from both n and n′ (i.e. it “crosses the diagram”).

Definition

The rank of a partition is the number of transversal blocks it has. α = β = rank(α) = 1, rank(β) = 2.

Michael Torpey (University of St Andrews) Diagram Semigroups 2017-06-07 5 / 8

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Attributes of partitions

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Attributes of partitions

Definition

The domain (resp. codomain) of a partition α ∈ Pn is the set of points i ∈ n (resp. i′ ∈ n′) which lie in transversal blocks.

Michael Torpey (University of St Andrews) Diagram Semigroups 2017-06-07 6 / 8

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Attributes of partitions

Definition

The domain (resp. codomain) of a partition α ∈ Pn is the set of points i ∈ n (resp. i′ ∈ n′) which lie in transversal blocks.

Definition

The kernel (resp. cokernel) of a partition α ∈ Pn is the restriction of α to n (resp. n′).

Michael Torpey (University of St Andrews) Diagram Semigroups 2017-06-07 6 / 8

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Attributes of partitions

Definition

The domain (resp. codomain) of a partition α ∈ Pn is the set of points i ∈ n (resp. i′ ∈ n′) which lie in transversal blocks.

Definition

The kernel (resp. cokernel) of a partition α ∈ Pn is the restriction of α to n (resp. n′). α =

Michael Torpey (University of St Andrews) Diagram Semigroups 2017-06-07 6 / 8

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Attributes of partitions

Definition

The domain (resp. codomain) of a partition α ∈ Pn is the set of points i ∈ n (resp. i′ ∈ n′) which lie in transversal blocks.

Definition

The kernel (resp. cokernel) of a partition α ∈ Pn is the restriction of α to n (resp. n′). α = dom α = {1, 3, 4},

Michael Torpey (University of St Andrews) Diagram Semigroups 2017-06-07 6 / 8

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Attributes of partitions

Definition

The domain (resp. codomain) of a partition α ∈ Pn is the set of points i ∈ n (resp. i′ ∈ n′) which lie in transversal blocks.

Definition

The kernel (resp. cokernel) of a partition α ∈ Pn is the restriction of α to n (resp. n′). α = dom α = {1, 3, 4}, codom α = {1′, 2′},

Michael Torpey (University of St Andrews) Diagram Semigroups 2017-06-07 6 / 8

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Attributes of partitions

Definition

The domain (resp. codomain) of a partition α ∈ Pn is the set of points i ∈ n (resp. i′ ∈ n′) which lie in transversal blocks.

Definition

The kernel (resp. cokernel) of a partition α ∈ Pn is the restriction of α to n (resp. n′). α = dom α = {1, 3, 4}, codom α = {1′, 2′}, ker α =

{1, 3, 4}, {2}, {5} ,

Michael Torpey (University of St Andrews) Diagram Semigroups 2017-06-07 6 / 8

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Attributes of partitions

Definition

The domain (resp. codomain) of a partition α ∈ Pn is the set of points i ∈ n (resp. i′ ∈ n′) which lie in transversal blocks.

Definition

The kernel (resp. cokernel) of a partition α ∈ Pn is the restriction of α to n (resp. n′). α = dom α = {1, 3, 4}, codom α = {1′, 2′}, ker α =

{1, 3, 4}, {2}, {5} ,

coker α =

{1′, 2′}, {3′, 4′}, {5′}

Michael Torpey (University of St Andrews)

Diagram Semigroups 2017-06-07 6 / 8

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Submonoids of Pn

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Submonoids of Pn

Tn – the full transformation monoid embeds as seen

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Submonoids of Pn

Tn – the full transformation monoid embeds as seen On ≤ In ≤ PT n embed similarly

Michael Torpey (University of St Andrews) Diagram Semigroups 2017-06-07 7 / 8

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Submonoids of Pn

Tn – the full transformation monoid embeds as seen On ≤ In ≤ PT n embed similarly Bn – the Brauer monoid – each block has size 2

Michael Torpey (University of St Andrews) Diagram Semigroups 2017-06-07 7 / 8

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Submonoids of Pn

Tn – the full transformation monoid embeds as seen On ≤ In ≤ PT n embed similarly Bn – the Brauer monoid – each block has size 2 PBn – the partial Brauer monoid – each block has size 1 or 2

Michael Torpey (University of St Andrews) Diagram Semigroups 2017-06-07 7 / 8

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Submonoids of Pn

Tn – the full transformation monoid embeds as seen On ≤ In ≤ PT n embed similarly Bn – the Brauer monoid – each block has size 2 PBn – the partial Brauer monoid – each block has size 1 or 2 PPn – the planar partition monoid – diagram is planar

Michael Torpey (University of St Andrews) Diagram Semigroups 2017-06-07 7 / 8

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Submonoids of Pn

Tn – the full transformation monoid embeds as seen On ≤ In ≤ PT n embed similarly Bn – the Brauer monoid – each block has size 2 PBn – the partial Brauer monoid – each block has size 1 or 2 PPn – the planar partition monoid – diagram is planar Jn – the Jones monoid – diagram is planar; block size 2

Michael Torpey (University of St Andrews) Diagram Semigroups 2017-06-07 7 / 8

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Submonoids of Pn

Tn – the full transformation monoid embeds as seen On ≤ In ≤ PT n embed similarly Bn – the Brauer monoid – each block has size 2 PBn – the partial Brauer monoid – each block has size 1 or 2 PPn – the planar partition monoid – diagram is planar Jn – the Jones monoid – diagram is planar; block size 2 Mn – the Motzkin monoid – diagram is planar; block size 1 or 2 [2]

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Howie, J.M., Fundamentals of Semigroup Theory, Oxford Science Publications, 1995, 1.1, 1.5 & 1.8, 7-35. James East, Attila Egri-Nagy, Andrew R. Francis, James D. Mitchell, Finite Diagram Semigroups: Extending the Computational Horizon, https://arxiv.org/abs/1502.07150

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