Algebras of incidence structures: representing regular double - PowerPoint PPT Presentation

Algebras of incidence structures: representing regular double p-algebras Christopher Taylor La Trobe University AustMS 2015 Chris Taylor Algebras of incidence structures AustMS 2015 1 / 25

Acknowledgements Thanks to the AustMS Student Support Scheme for providing additional funding to help attend the conference. Chris Taylor Algebras of incidence structures AustMS 2015 2 / 25

Boolean lattices P ( { 1 , 2 , 3 } ) = { ∅ , { 1 } , { 2 } , { 3 } , { 1 , 2 } , { 1 , 3 } , { 2 , 3 } , { 1 , 2 , 3 }} Chris Taylor Algebras of incidence structures AustMS 2015 3 / 25

Boolean lattices P ( { 1 , 2 , 3 } ) = { ∅ , { 1 } , { 2 } , { 3 } , { 1 , 2 } , { 1 , 3 } , { 2 , 3 } , { 1 , 2 , 3 }} ∅ Chris Taylor Algebras of incidence structures AustMS 2015 3 / 25

Boolean lattices P ( { 1 , 2 , 3 } ) = { ∅ , { 1 } , { 2 } , { 3 } , { 1 , 2 } , { 1 , 3 } , { 2 , 3 } , { 1 , 2 , 3 }} { 1 } ∅ Chris Taylor Algebras of incidence structures AustMS 2015 3 / 25

Boolean lattices P ( { 1 , 2 , 3 } ) = { ∅ , { 1 } , { 2 } , { 3 } , { 1 , 2 } , { 1 , 3 } , { 2 , 3 } , { 1 , 2 , 3 }} { 1 } { 2 } ∅ Chris Taylor Algebras of incidence structures AustMS 2015 3 / 25

Boolean lattices P ( { 1 , 2 , 3 } ) = { ∅ , { 1 } , { 2 } , { 3 } , { 1 , 2 } , { 1 , 3 } , { 2 , 3 } , { 1 , 2 , 3 }} { 1 } { 3 } { 2 } ∅ Chris Taylor Algebras of incidence structures AustMS 2015 3 / 25

Boolean lattices P ( { 1 , 2 , 3 } ) = { ∅ , { 1 } , { 2 } , { 3 } , { 1 , 2 } , { 1 , 3 } , { 2 , 3 } , { 1 , 2 , 3 }} { 1 , 2 } { 1 } { 3 } { 2 } ∅ Chris Taylor Algebras of incidence structures AustMS 2015 3 / 25

Boolean lattices P ( { 1 , 2 , 3 } ) = { ∅ , { 1 } , { 2 } , { 3 } , { 1 , 2 } , { 1 , 3 } , { 2 , 3 } , { 1 , 2 , 3 }} { 1 , 3 } { 1 , 2 } { 1 } { 3 } { 2 } ∅ Chris Taylor Algebras of incidence structures AustMS 2015 3 / 25

Boolean lattices P ( { 1 , 2 , 3 } ) = { ∅ , { 1 } , { 2 } , { 3 } , { 1 , 2 } , { 1 , 3 } , { 2 , 3 } , { 1 , 2 , 3 }} { 1 , 3 } { 1 , 2 } { 2 , 3 } { 1 } { 3 } { 2 } ∅ Chris Taylor Algebras of incidence structures AustMS 2015 3 / 25

Boolean lattices P ( { 1 , 2 , 3 } ) = { ∅ , { 1 } , { 2 } , { 3 } , { 1 , 2 } , { 1 , 3 } , { 2 , 3 } , { 1 , 2 , 3 }} { 1 , 2 , 3 } { 1 , 3 } { 1 , 2 } { 2 , 3 } { 1 } { 3 } { 2 } ∅ Chris Taylor Algebras of incidence structures AustMS 2015 3 / 25

Boolean lattices Definition Boolean lattice : a bounded distributive lattice B = � B ; ∨ , ∧ , 0 , 1 � such that every x ∈ B has a (unique) complement. Chris Taylor Algebras of incidence structures AustMS 2015 4 / 25

Boolean lattices Definition Boolean lattice : a bounded distributive lattice B = � B ; ∨ , ∧ , 0 , 1 � such that every x ∈ B has a (unique) complement. Theorem Let L be a finite lattice. Then the following are equivalent. L is a boolean lattice, 1 L ∼ = P ( B ) for some finite set B, 2 = 2 n for some n ≥ 0 . L ∼ 3 Chris Taylor Algebras of incidence structures AustMS 2015 4 / 25

Some other classifications Birkhoff’s duality for finite distributive lattices Stone’s duality for boolean algebras Priestley’s duality for bounded distributive lattices Every finite cyclic group is isomorphic to Z n for some n ∈ ω Every finite abelian group is isomorphic to � n i = 0 Z q i where each q i is a power of a prime Chris Taylor Algebras of incidence structures AustMS 2015 5 / 25

Graphs A graph: Chris Taylor Algebras of incidence structures AustMS 2015 6 / 25

Graphs A graph: A subgraph: Chris Taylor Algebras of incidence structures AustMS 2015 6 / 25

Graphs A graph: A subgraph: Chris Taylor Algebras of incidence structures AustMS 2015 6 / 25

Graphs A graph: A subgraph: Chris Taylor Algebras of incidence structures AustMS 2015 6 / 25

Graphs A graph: A subgraph: Chris Taylor Algebras of incidence structures AustMS 2015 6 / 25

Graphs A graph: A subgraph: Chris Taylor Algebras of incidence structures AustMS 2015 6 / 25

Graphs A graph: A subgraph: Chris Taylor Algebras of incidence structures AustMS 2015 6 / 25

Graphs A graph: A subgraph: Chris Taylor Algebras of incidence structures AustMS 2015 6 / 25

Graphs A graph: A subgraph: Chris Taylor Algebras of incidence structures AustMS 2015 6 / 25

Graphs A graph: A subgraph: Chris Taylor Algebras of incidence structures AustMS 2015 6 / 25

Graphs A graph: A subgraph: Chris Taylor Algebras of incidence structures AustMS 2015 6 / 25

Graphs A graph: A subgraph: Chris Taylor Algebras of incidence structures AustMS 2015 6 / 25

Graphs A graph: A subgraph: Chris Taylor Algebras of incidence structures AustMS 2015 6 / 25

Graphs A graph: A subgraph: Chris Taylor Algebras of incidence structures AustMS 2015 6 / 25

The lattice of subgraphs Let G = � V , E � be a graph. The set of all subgraphs of G induces a bounded distributive lattice, which we will call S ( G ) , where � V 1 , E 1 � ∨ � V 2 , E 2 � = � V 1 ∪ V 2 , E 1 ∪ E 2 � � V 1 , E 1 � ∧ � V 2 , E 2 � = � V 1 ∩ V 2 , E 1 ∩ E 2 � . Note that we permit the empty graph. Chris Taylor Algebras of incidence structures AustMS 2015 7 / 25

Graph complements Chris Taylor Algebras of incidence structures AustMS 2015 8 / 25

Graph complements Chris Taylor Algebras of incidence structures AustMS 2015 8 / 25

Graph complements Complement ֒ → Chris Taylor Algebras of incidence structures AustMS 2015 8 / 25

Graph complements Complement ֒ → Chris Taylor Algebras of incidence structures AustMS 2015 9 / 25

Graph complements ֒ → Chris Taylor Algebras of incidence structures AustMS 2015 9 / 25

Graph complements ֒ → Chris Taylor Algebras of incidence structures AustMS 2015 10 / 25

Graph complements ֒ → Chris Taylor Algebras of incidence structures AustMS 2015 10 / 25

Pseudocomplementation Let L be a lattice and let x ∈ L . Then x has a pseudocomplement if there exists a largest element x ∗ ∈ L such that x ∧ x ∗ = 0. Chris Taylor Algebras of incidence structures AustMS 2015 11 / 25

Pseudocomplementation Let L be a lattice and let x ∈ L . Then x has a pseudocomplement if there exists a largest element x ∗ ∈ L such that x ∧ x ∗ = 0. Example: The lattice of open sets of a topological space X . If U is an open set, then U ∗ = int ( X \ U ) . Chris Taylor Algebras of incidence structures AustMS 2015 11 / 25

Pseudocomplementation Let L be a lattice and let x ∈ L . Then x has a pseudocomplement if there exists a largest element x ∗ ∈ L such that x ∧ x ∗ = 0. Example: The lattice of open sets of a topological space X . If U is an open set, then U ∗ = int ( X \ U ) . Let L be a lattice and let x ∈ L . Then x has a dual pseudocomplement if there exists a smallest element x + ∈ L such that x ∨ x + = 1. Chris Taylor Algebras of incidence structures AustMS 2015 11 / 25

Pseudocomplementation Let L be a lattice and let x ∈ L . Then x has a pseudocomplement if there exists a largest element x ∗ ∈ L such that x ∧ x ∗ = 0. Example: The lattice of open sets of a topological space X . If U is an open set, then U ∗ = int ( X \ U ) . Let L be a lattice and let x ∈ L . Then x has a dual pseudocomplement if there exists a smallest element x + ∈ L such that x ∨ x + = 1. Example: The lattice of closed sets of a topological space X . If C is a closed set, then U + = cl ( X \ C ) . Chris Taylor Algebras of incidence structures AustMS 2015 11 / 25

Pseudocomplementation Let L be a lattice and let x ∈ L . Then x has a pseudocomplement if there exists a largest element x ∗ ∈ L such that x ∧ x ∗ = 0. Example: The lattice of open sets of a topological space X . If U is an open set, then U ∗ = int ( X \ U ) . Let L be a lattice and let x ∈ L . Then x has a dual pseudocomplement if there exists a smallest element x + ∈ L such that x ∨ x + = 1. Example: The lattice of closed sets of a topological space X . If C is a closed set, then U + = cl ( X \ C ) . Definition An algebra A = � A ; ∨ , ∧ , 0 , 1 , ∗ , + � is a double p-algebra if � A ; ∨ , ∧ , 0 , 1 � is a bounded lattice, and ∗ and + are the pseudocomplement and dual pseudocomplement respectively. Chris Taylor Algebras of incidence structures AustMS 2015 11 / 25

The algebra of subgraphs Pseudocomplement Take the set complement of the subgraph and abandon the extra edges. Chris Taylor Algebras of incidence structures AustMS 2015 12 / 25

The algebra of subgraphs Pseudocomplement Take the set complement of the subgraph and abandon the extra edges. Dual pseudocomplement Just add the missing vertices back Chris Taylor Algebras of incidence structures AustMS 2015 12 / 25

The algebra of subgraphs Pseudocomplement Take the set complement of the subgraph and abandon the extra edges. Dual pseudocomplement Just add the missing vertices back Formally, for a graph G = � V , E � and a subgraph H = � V ′ , E ′ � : H ∗ = � V \ V ′ , { e ∈ E \ E ′ | ( ∀ x ∈ e ) x ∈ V \ V ′ }� H + = � V \ V ′ ∪ { v ∈ V | ( ∃ e ∈ E \ E ′ ) v ∈ e } , E \ E ′ � . Chris Taylor Algebras of incidence structures AustMS 2015 12 / 25