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

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 Andrews) Diagram Semigroups 2017-06-07 2 / 8

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

Transformations Definition A transformation on a set X is any function τ : X → X Assume X = n = { 1 , 2 , . . . , n } , and write T X as T n . Michael Torpey (University of St Andrews) Diagram Semigroups 2017-06-07 2 / 8

Transformations Definition A transformation on a set X is any function τ : X → X Assume X = n = { 1 , 2 , . . . , n } , and write T X as T n . Theorem (Cayley for semigroups) Every semigroup S is isomorphic to a subsemigroup of the full transformation semigroup T S . [1, p.7] Michael Torpey (University of St Andrews) Diagram Semigroups 2017-06-07 2 / 8

Transformations Definition A transformation on a set X is any function τ : X → X Assume X = n = { 1 , 2 , . . . , n } , and write T X as T n . Theorem (Cayley for semigroups) Every semigroup S is isomorphic to a subsemigroup of the full transformation semigroup T S . [1, p.7] � � � � 1 2 3 4 5 1 2 3 4 5 α = , β = 1 3 1 5 5 3 1 3 3 5 Michael Torpey (University of St Andrews) Diagram Semigroups 2017-06-07 2 / 8

Transformations Definition A transformation on a set X is any function τ : X → X Assume X = n = { 1 , 2 , . . . , n } , and write T X as T n . Theorem (Cayley for semigroups) Every semigroup S is isomorphic to a subsemigroup of the full transformation semigroup T S . [1, p.7] � � � � 1 2 3 4 5 1 2 3 4 5 α = , β = 1 3 1 5 5 3 1 3 3 5 α = β = , , Michael Torpey (University of St Andrews) Diagram Semigroups 2017-06-07 2 / 8

Transformations Definition A transformation on a set X is any function τ : X → X Assume X = n = { 1 , 2 , . . . , n } , and write T X as T n . Theorem (Cayley for semigroups) Every semigroup S is isomorphic to a subsemigroup of the full transformation semigroup T S . [1, p.7] � � � � 1 2 3 4 5 1 2 3 4 5 α = , β = 1 3 1 5 5 3 1 3 3 5 α = β = αβ = , , Michael Torpey (University of St Andrews) Diagram Semigroups 2017-06-07 2 / 8

Transformations Definition A transformation on a set X is any function τ : X → X Assume X = n = { 1 , 2 , . . . , n } , and write T X as T n . Theorem (Cayley for semigroups) Every semigroup S is isomorphic to a subsemigroup of the full transformation semigroup T S . [1, p.7] � � � � 1 2 3 4 5 1 2 3 4 5 α = , β = 1 3 1 5 5 3 1 3 3 5 α = β = αβ = = , , Michael Torpey (University of St Andrews) Diagram Semigroups 2017-06-07 2 / 8

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

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

Partitions P n – the (bi)partition monoid Michael Torpey (University of St Andrews) Diagram Semigroups 2017-06-07 3 / 8

Partitions P n – the (bi)partition monoid α = , Michael Torpey (University of St Andrews) Diagram Semigroups 2017-06-07 3 / 8

Partitions P n – the (bi)partition monoid α = , Michael Torpey (University of St Andrews) Diagram Semigroups 2017-06-07 3 / 8

Partitions P n – the (bi)partition monoid α = , β = Michael Torpey (University of St Andrews) Diagram Semigroups 2017-06-07 3 / 8

Partitions P n – 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

Partitions P n – 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

Partitions P n – 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

Partitions P n – 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

Partitions P n – 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

Partitions P n – 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

Partitions P n – 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

Partitions P n – 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

How big is the partition monoid? Michael Torpey (University of St Andrews) Diagram Semigroups 2017-06-07 4 / 8

How big is the partition monoid? P n consists of all partitions of 2 n points. Michael Torpey (University of St Andrews) Diagram Semigroups 2017-06-07 4 / 8

How big is the partition monoid? P n consists of all partitions of 2 n points. Its size is the Bell number B 2 n . Michael Torpey (University of St Andrews) Diagram Semigroups 2017-06-07 4 / 8

How big is the partition monoid? P n consists of all partitions of 2 n points. Its size is the Bell number B 2 n . n 1 2 3 4 5 6 7 . . . |P n | 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

How big is the partition monoid? P n consists of all partitions of 2 n points. Its size is the Bell number B 2 n . n 1 2 3 4 5 6 7 . . . |P n | 2 15 203 4,140 115,975 4,213,597 190,899,322 . . . Table: Sizes of partition monoids |P 10 | ≈ 5 . 2 × 10 13 . Michael Torpey (University of St Andrews) Diagram Semigroups 2017-06-07 4 / 8

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

Attributes of partitions Definition A block in a partition α ∈ P n 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

Attributes of partitions Definition A block in a partition α ∈ P n 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

Attributes of partitions Definition A block in a partition α ∈ P n 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

Attributes of partitions Definition A block in a partition α ∈ P n 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

Attributes of partitions Definition A block in a partition α ∈ P n 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

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

Attributes of partitions Definition The domain ( resp. codomain ) of a partition α ∈ P n 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