Pseudovarieties generated by Brauer type monoids K. Auinger NSAC - PowerPoint PPT Presentation

Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product Pseudovarieties generated by Brauer type monoids K. Auinger NSAC 2013 K. Auinger Pseudovarieties generated by Brauer type monoids

Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product all monoids (semigroups) in this talk are finite K. Auinger Pseudovarieties generated by Brauer type monoids

Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product all monoids (semigroups) in this talk are finite B n = all partitions of { 1 , . . . , n , 1 ′ , . . . , n ′ } into blocks of size 2 K. Auinger Pseudovarieties generated by Brauer type monoids

Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product all monoids (semigroups) in this talk are finite B n = all partitions of { 1 , . . . , n , 1 ′ , . . . , n ′ } into blocks of size 2 members of B n are diagrams like this: 1 1 ′ 7 7 ′ K. Auinger Pseudovarieties generated by Brauer type monoids

Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product Composition of diagrams K. Auinger Pseudovarieties generated by Brauer type monoids

Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product Composition of diagrams K. Auinger Pseudovarieties generated by Brauer type monoids

Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product Composition of diagrams K. Auinger Pseudovarieties generated by Brauer type monoids

Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product Composition of diagrams = K. Auinger Pseudovarieties generated by Brauer type monoids

Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product Composition of diagrams = K. Auinger Pseudovarieties generated by Brauer type monoids

Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product Composition of diagrams = = K. Auinger Pseudovarieties generated by Brauer type monoids

Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product Composition of diagrams = = composition of diagrams defines a monoid structure on B n , the Brauer monoid K. Auinger Pseudovarieties generated by Brauer type monoids

Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product J n = all elements of B n whose diagrams can be drawn without intersecting lines: K. Auinger Pseudovarieties generated by Brauer type monoids

Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product J n = all elements of B n whose diagrams can be drawn without intersecting lines: K. Auinger Pseudovarieties generated by Brauer type monoids

Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product J n = all elements of B n whose diagrams can be drawn without intersecting lines: J n is closed under composition of diagrams, is called the Jones monoid or the Temperley–Lieb monoid K. Auinger Pseudovarieties generated by Brauer type monoids

Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product Question K. Auinger Pseudovarieties generated by Brauer type monoids

Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product Question What is pvar { J n | n ∈ N } ? K. Auinger Pseudovarieties generated by Brauer type monoids

Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product Question What is pvar { J n | n ∈ N } ? What is pvar { B n | n ∈ N } ? K. Auinger Pseudovarieties generated by Brauer type monoids

Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product Question What is pvar { J n | n ∈ N } ? What is pvar { B n | n ∈ N } ? K. Auinger Pseudovarieties generated by Brauer type monoids

Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product Question What is pvar { J n | n ∈ N } ? What is pvar { B n | n ∈ N } ? Clearly, K. Auinger Pseudovarieties generated by Brauer type monoids

Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product Question What is pvar { J n | n ∈ N } ? What is pvar { B n | n ∈ N } ? Clearly, J n × J m ֒ → J n + m K. Auinger Pseudovarieties generated by Brauer type monoids

Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product Question What is pvar { J n | n ∈ N } ? What is pvar { B n | n ∈ N } ? Clearly, J n × J m ֒ → J n + m B n × B m ֒ → B n + m . K. Auinger Pseudovarieties generated by Brauer type monoids

Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product Question What is pvar { J n | n ∈ N } ? What is pvar { B n | n ∈ N } ? Clearly, J n × J m ֒ → J n + m B n × B m ֒ → B n + m . K. Auinger Pseudovarieties generated by Brauer type monoids

Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product Question What is pvar { J n | n ∈ N } ? What is pvar { B n | n ∈ N } ? Clearly, J n × J m ֒ → J n + m B n × B m ֒ → B n + m . Hence, K. Auinger Pseudovarieties generated by Brauer type monoids

Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product Question What is pvar { J n | n ∈ N } ? What is pvar { B n | n ∈ N } ? Clearly, J n × J m ֒ → J n + m B n × B m ֒ → B n + m . Hence, pvar { J n | n ∈ N } = the divisors of all J n K. Auinger Pseudovarieties generated by Brauer type monoids

Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product Question What is pvar { J n | n ∈ N } ? What is pvar { B n | n ∈ N } ? Clearly, J n × J m ֒ → J n + m B n × B m ֒ → B n + m . Hence, pvar { J n | n ∈ N } = the divisors of all J n pvar { B n | n ∈ N } = the divisors of all B n . K. Auinger Pseudovarieties generated by Brauer type monoids

Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product A = all aperiodic monoids K. Auinger Pseudovarieties generated by Brauer type monoids

Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product A = all aperiodic monoids M = all monoids. K. Auinger Pseudovarieties generated by Brauer type monoids

Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product A = all aperiodic monoids M = all monoids. Theorem K. Auinger Pseudovarieties generated by Brauer type monoids

Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product A = all aperiodic monoids M = all monoids. Theorem pvar { J n | n ∈ N } = A K. Auinger Pseudovarieties generated by Brauer type monoids

Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product A = all aperiodic monoids M = all monoids. Theorem pvar { J n | n ∈ N } = A pvar { B n | n ∈ N } = M . K. Auinger Pseudovarieties generated by Brauer type monoids

Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product A = all aperiodic monoids M = all monoids. Theorem pvar { J n | n ∈ N } = A pvar { B n | n ∈ N } = M . K. Auinger Pseudovarieties generated by Brauer type monoids

Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product A = all aperiodic monoids M = all monoids. Theorem pvar { J n | n ∈ N } = A pvar { B n | n ∈ N } = M . In other words, K. Auinger Pseudovarieties generated by Brauer type monoids

Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product A = all aperiodic monoids M = all monoids. Theorem pvar { J n | n ∈ N } = A pvar { B n | n ∈ N } = M . In other words, each aperiodic monoid divides some J n , K. Auinger Pseudovarieties generated by Brauer type monoids