Preserving meets in meet-completions Robert Egrot, with thanks to - PowerPoint PPT Presentation

Preserving meets in meet-completions Robert Egrot, with thanks to Robin Hirsch February 16, 2012 Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 1 / 23

meet-completions e : P → L Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 2 / 23

meet-completions e : P → L P is a poset, Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 2 / 23

meet-completions e : P → L P is a poset, L is a complete lattice, Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 2 / 23

meet-completions e : P → L P is a poset, L is a complete lattice, e is a map with p ≤ q ⇐ ⇒ e ( p ) ≤ e ( q ) ( e is an embedding), and Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 2 / 23

meet-completions e : P → L P is a poset, L is a complete lattice, e is a map with p ≤ q ⇐ ⇒ e ( p ) ≤ e ( q ) ( e is an embedding), and whenever q 1 , q 2 ∈ L and q 1 �≤ q 2 there is p ∈ P with e ( p ) ≥ q 2 and e ( p ) �≥ q 1 ( e [ P ] is meet-dense in L ). Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 2 / 23

meet-completions e : P → L P is a poset, L is a complete lattice, e is a map with p ≤ q ⇐ ⇒ e ( p ) ≤ e ( q ) ( e is an embedding), and whenever q 1 , q 2 ∈ L and q 1 �≤ q 2 there is p ∈ P with e ( p ) ≥ q 2 and e ( p ) �≥ q 1 ( e [ P ] is meet-dense in L ). We say e : P → L is a meet-completion of P . Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 2 / 23

Examples of meet-completions Example P * is the complete lattice of upsets of P (including ∅ ) ordered by reverse inclusion, ι : P → P * , p �→ p ↑ . Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 3 / 23

Examples of meet-completions Example P * is the complete lattice of upsets of P (including ∅ ) ordered by reverse inclusion, ι : P → P * , p �→ p ↑ . Example The MacNeille completion DM ( P ) (this can be constructed as the set of all normal filters ordered by reverse inclusion, with the embedding ι ′ : p �→ p ↑ ). Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 3 / 23

Examples of meet-completions Example P * is the complete lattice of upsets of P (including ∅ ) ordered by reverse inclusion, ι : P → P * , p �→ p ↑ . Example The MacNeille completion DM ( P ) (this can be constructed as the set of all normal filters ordered by reverse inclusion, with the embedding ι ′ : p �→ p ↑ ). ι ′ preserves all meets and joins that are defined in P (this defines DM ( P ) up to isomorphism), while ι preserves all the existing joins but destroys all existing meets. Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 3 / 23

meet-completions and closure operators Γ: P → P Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 4 / 23

meet-completions and closure operators Γ: P → P p ≤ Γ( p ) for all p ∈ P , Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 4 / 23

meet-completions and closure operators Γ: P → P p ≤ Γ( p ) for all p ∈ P , p ≤ q = ⇒ Γ( p ) ≤ Γ( q ) for all p , q ∈ P , and Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 4 / 23

meet-completions and closure operators Γ: P → P p ≤ Γ( p ) for all p ∈ P , p ≤ q = ⇒ Γ( p ) ≤ Γ( q ) for all p , q ∈ P , and Γ(Γ( p )) = Γ( p ) for all p ∈ P . Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 4 / 23

meet-completions and closure operators Γ: P → P p ≤ Γ( p ) for all p ∈ P , p ≤ q = ⇒ Γ( p ) ≤ Γ( q ) for all p , q ∈ P , and Γ(Γ( p )) = Γ( p ) for all p ∈ P . Γ is a closure operator on P . Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 4 / 23

standard closure operators A closure operator Γ: P * → P * is standard when Γ( p ↑ ) = p ↑ for all p ∈ P . Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 5 / 23

standard closure operators A closure operator Γ: P * → P * is standard when Γ( p ↑ ) = p ↑ for all p ∈ P . Given a meet-completion e : P → Q the map Γ e : S �→ { p ∈ P : e ( p ) ≥ � e [ S ] } defines a standard closure operator on P * δ . Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 5 / 23

standard closure operators A closure operator Γ: P * → P * is standard when Γ( p ↑ ) = p ↑ for all p ∈ P . Given a meet-completion e : P → Q the map Γ e : S �→ { p ∈ P : e ( p ) ≥ � e [ S ] } defines a standard closure operator on P * δ . If Γ is a standard closure operator on P * δ then the Γ closed sets and the embedding e Γ : p → p ↑ defines a meet-completion of P . Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 5 / 23

� � � standard closure operators A closure operator Γ: P * → P * is standard when Γ( p ↑ ) = p ↑ for all p ∈ P . Given a meet-completion e : P → Q the map Γ e : S �→ { p ∈ P : e ( p ) ≥ � e [ S ] } defines a standard closure operator on P * δ . If Γ is a standard closure operator on P * δ then the Γ closed sets and the embedding e Γ : p → p ↑ defines a meet-completion of P . Theorem If e : P → Q is a meet-completion then there is a unique isomorphism such that the following commutes: e Q P � �������� e Γ e ∼ = Γ e [ P * ] Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 5 / 23

M P and S P define M P to be the complete lattice of meet-completions of P (up to isomorphism, and ordered by inclusion lifting the identity on P ) Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 6 / 23

M P and S P define M P to be the complete lattice of meet-completions of P (up to isomorphism, and ordered by inclusion lifting the identity on P ) define S P to be the complete lattice of standard closure operators on P * δ (ordered pointwise, i.e. ⇒ Γ 1 ( S ) ⊆ Γ 2 ( S ) for all S ∈ P * ) Γ 1 ≤ Γ 2 ⇐ ⇒ Γ 1 ( S ) ≤ Γ 2 ( S ) ⇐ Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 6 / 23

M P and S P define M P to be the complete lattice of meet-completions of P (up to isomorphism, and ordered by inclusion lifting the identity on P ) define S P to be the complete lattice of standard closure operators on P * δ (ordered pointwise, i.e. ⇒ Γ 1 ( S ) ⊆ Γ 2 ( S ) for all S ∈ P * ) Γ 1 ≤ Γ 2 ⇐ ⇒ Γ 1 ( S ) ≤ Γ 2 ( S ) ⇐ Theorem M P ∼ = δ S P . Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 6 / 23

Meet-completions preserve all joins Proposition Let e : P → Q be a meet-completion of P, then for all S , T ⊆ P, � S = � T = ⇒ � e [ S ] = � e [ T ] . Conversely, if either � S or � T exist in P then � e [ S ] = � e [ T ] = ⇒ they both exist and are equal. Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 7 / 23

Meet-completions preserve all joins Proposition Let e : P → Q be a meet-completion of P, then for all S , T ⊆ P, � S = � T = ⇒ � e [ S ] = � e [ T ] . Conversely, if either � S or � T exist in P then � e [ S ] = � e [ T ] = ⇒ they both exist and are equal. Corollary If e : P → Q is a meet-completion then e ( � S ) = � e [ S ] for all S ⊆ P where � S is defined. Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 7 / 23

Meet-completions preserve all joins Proposition Let e : P → Q be a meet-completion of P, then for all S , T ⊆ P, � S = � T = ⇒ � e [ S ] = � e [ T ] . Conversely, if either � S or � T exist in P then � e [ S ] = � e [ T ] = ⇒ they both exist and are equal. Corollary If e : P → Q is a meet-completion then e ( � S ) = � e [ S ] for all S ⊆ P where � S is defined. Proof. Take T = { � S } . Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 7 / 23

What meets can be preserved? Definition ( S -regular) Given a poset P and S ⊆ P * such that � S is defined in P for all S ∈ S , we say a meet-completion e : P → Q is S -regular if e ( � S ) = � e [ S ] for all S ∈ S . Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 8 / 23

What meets can be preserved? Definition ( S -regular) Given a poset P and S ⊆ P * such that � S is defined in P for all S ∈ S , we say a meet-completion e : P → Q is S -regular if e ( � S ) = � e [ S ] for all S ∈ S . Question Given S ⊆ P * , when does an S -regular meet-completion exist? Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 8 / 23