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

Algebras of incidence structures: representing regular double p-algebras Christopher Taylor La Trobe University Victorian Algebra Conference 2015 Chris Taylor Algebras of incidence structures VAC 2015 1 / 24

Chris Taylor Algebras of incidence structures VAC 2015 1 / 24

Boolean lattices 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 L ∼ = 2 n for some n ≥ 0 . 3 Chris Taylor Algebras of incidence structures VAC 2015 2 / 24

Boolean lattices 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 L ∼ = 2 n for some n ≥ 0 . 3 Theorem Let B be a boolean lattice. Then the following are equivalent. B ∼ = P ( X ) for some set X. 1 B is complete and atomic. 2 B is complete and completely distributive. 3 Chris Taylor Algebras of incidence structures VAC 2015 2 / 24

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 VAC 2015 3 / 24

Graphs A graph: Chris Taylor Algebras of incidence structures VAC 2015 4 / 24

Graphs A graph: A subgraph: Chris Taylor Algebras of incidence structures VAC 2015 4 / 24

Graphs A graph: A subgraph: Chris Taylor Algebras of incidence structures VAC 2015 4 / 24

Graphs A graph: A subgraph: Chris Taylor Algebras of incidence structures VAC 2015 4 / 24

Graphs A graph: A subgraph: Chris Taylor Algebras of incidence structures VAC 2015 4 / 24

Graphs A graph: A subgraph: Chris Taylor Algebras of incidence structures VAC 2015 4 / 24

Graphs A graph: A subgraph: Chris Taylor Algebras of incidence structures VAC 2015 4 / 24

Graphs A graph: A subgraph: Chris Taylor Algebras of incidence structures VAC 2015 4 / 24

Graphs A graph: A subgraph: Chris Taylor Algebras of incidence structures VAC 2015 4 / 24

Graphs A graph: A subgraph: Chris Taylor Algebras of incidence structures VAC 2015 4 / 24

Graphs A graph: A subgraph: Chris Taylor Algebras of incidence structures VAC 2015 4 / 24

Graphs A graph: A subgraph: Chris Taylor Algebras of incidence structures VAC 2015 4 / 24

Graphs A graph: A subgraph: Chris Taylor Algebras of incidence structures VAC 2015 4 / 24

Graphs A graph: A subgraph: Chris Taylor Algebras of incidence structures VAC 2015 4 / 24

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 VAC 2015 5 / 24

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. Theorem (Reyes & Zolfaghari, 1996) Let G be a graph. Then S ( G ) naturally forms a double-Heyting algebra. Chris Taylor Algebras of incidence structures VAC 2015 5 / 24

Graph complements Chris Taylor Algebras of incidence structures VAC 2015 6 / 24

Graph complements Chris Taylor Algebras of incidence structures VAC 2015 6 / 24

Graph complements Complement ֒ → Chris Taylor Algebras of incidence structures VAC 2015 6 / 24

Graph complements Complement ֒ → Chris Taylor Algebras of incidence structures VAC 2015 7 / 24

Graph complements ֒ → Chris Taylor Algebras of incidence structures VAC 2015 7 / 24

Graph complements ֒ → Chris Taylor Algebras of incidence structures VAC 2015 8 / 24

Graph complements ֒ → Chris Taylor Algebras of incidence structures VAC 2015 8 / 24

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 VAC 2015 9 / 24

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 VAC 2015 9 / 24

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 VAC 2015 9 / 24

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 VAC 2015 9 / 24

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 VAC 2015 9 / 24

The algebra of subgraphs Pseudocomplement Take the set complement of the subgraph and abandon the extra edges. Formally, for a graph G = � V , E � and a subgraph H = � V ′ , E ′ � : H ∗ = � V \ V ′ , { e ∈ E \ E ′ | ( ∀ x ∈ e ) x ∈ V \ V ′ }� Chris Taylor Algebras of incidence structures VAC 2015 10 / 24

The algebra of subgraphs Pseudocomplement Take the set complement of the subgraph and abandon the extra edges. Formally, for a graph G = � V , E � and a subgraph H = � V ′ , E ′ � : H ∗ = � V \ V ′ , { e ∈ E \ E ′ | ( ∀ x ∈ e ) x ∈ V \ V ′ }� ֒ → Chris Taylor Algebras of incidence structures VAC 2015 10 / 24

The algebra of subgraphs Pseudocomplement Take the set complement of the subgraph and abandon the extra edges. Formally, for a graph G = � V , E � and a subgraph H = � V ′ , E ′ � : H ∗ = � V \ V ′ , { e ∈ E \ E ′ | ( ∀ x ∈ e ) x ∈ V \ V ′ }� ֒ → Chris Taylor Algebras of incidence structures VAC 2015 10 / 24

The algebra of subgraphs Dual pseudocomplement Just add the missing vertices back. Formally, for a graph G = � V , E � and a subgraph H = � V ′ , E ′ � : H + = � V \ V ′ ∪ { v ∈ V | ( ∃ e ∈ E \ E ′ ) v ∈ e } , E \ E ′ � Chris Taylor Algebras of incidence structures VAC 2015 11 / 24

The algebra of subgraphs Dual pseudocomplement Just add the missing vertices back. Formally, for a graph G = � V , E � and a subgraph H = � V ′ , E ′ � : H + = � V \ V ′ ∪ { v ∈ V | ( ∃ e ∈ E \ E ′ ) v ∈ e } , E \ E ′ � ֒ → Chris Taylor Algebras of incidence structures VAC 2015 11 / 24

The algebra of subgraphs Dual pseudocomplement Just add the missing vertices back. Formally, for a graph G = � V , E � and a subgraph H = � V ′ , E ′ � : H + = � V \ V ′ ∪ { v ∈ V | ( ∃ e ∈ E \ E ′ ) v ∈ e } , E \ E ′ � ֒ → Chris Taylor Algebras of incidence structures VAC 2015 11 / 24

Pseudocomplements are not bijective Boolean lattices: no two elements share a complement Double p-algebras: not true! Chris Taylor Algebras of incidence structures VAC 2015 12 / 24

Pseudocomplements are not bijective Boolean lattices: no two elements share a complement Double p-algebras: not true! ֒ → Chris Taylor Algebras of incidence structures VAC 2015 12 / 24