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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 q1, q2 ∈ L and q1 ≤ q2 there is p ∈ P with e(p) ≥ q2 and e(p) ≥ q1 (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 q1, q2 ∈ L and q1 ≤ q2 there is p ∈ P with e(p) ≥ q2 and e(p) ≥ q1 (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 Γe : S → {p ∈ P : e(p) ≥ e[S]} defines a standard closure operator

• n 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 Γe : S → {p ∈ P : e(p) ≥ e[S]} defines a standard closure operator

• n P*δ.

If Γ is a standard closure operator on P*δ then the Γ closed sets and the embedding eΓ : 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 Γe : S → {p ∈ P : e(p) ≥ e[S]} defines a standard closure operator

• n P*δ.

If Γ is a standard closure operator on P*δ then the Γ closed sets and the embedding eΓ : 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: P

e

• eΓe
• Q
• ∼

=

• Γe[P*]

Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 5 / 23

MP and SP

define MP 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

MP and SP

define MP to be the complete lattice of meet-completions of P (up to isomorphism, and ordered by inclusion lifting the identity on P) define SP to be the complete lattice of standard closure operators on P*δ (ordered pointwise, i.e. Γ1 ≤ Γ2 ⇐ ⇒ Γ1(S) ≤ Γ2(S) ⇐ ⇒ Γ1(S) ⊆ Γ2(S) for all S ∈ P*)

Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 6 / 23

MP and SP

define MP to be the complete lattice of meet-completions of P (up to isomorphism, and ordered by inclusion lifting the identity on P) define SP to be the complete lattice of standard closure operators on P*δ (ordered pointwise, i.e. Γ1 ≤ Γ2 ⇐ ⇒ Γ1(S) ≤ Γ2(S) ⇐ ⇒ Γ1(S) ⊆ Γ2(S) for all S ∈ P*)

Theorem

MP ∼ =δ SP.

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

What meets can be preserved? part II

Definition (regular)

Let P be a poset, then S ⊆ P* is regular if

Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 9 / 23

What meets can be preserved? part II

Definition (regular)

Let P be a poset, then S ⊆ P* is regular if

1 p↑ ∈ S for all p ∈ P, Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 9 / 23

What meets can be preserved? part II

Definition (regular)

Let P be a poset, then S ⊆ P* is regular if

1 p↑ ∈ S for all p ∈ P, 2

S exists in P for all S ∈ S , and

Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 9 / 23

What meets can be preserved? part II

Definition (regular)

Let P be a poset, then S ⊆ P* is regular if

1 p↑ ∈ S for all p ∈ P, 2

S exists in P for all S ∈ S , and

3 whenever T ∈ P* \ S , there is T ′ ∈ P* with T ⊆ T ′,

p < T = ⇒ p < T ′ for all p ∈ P, and for all S ∈ S , S ⊆ T ′ = ⇒ S ∈ T ′.

Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 9 / 23

What meets can be preserved? part II

Definition (regular)

Let P be a poset, then S ⊆ P* is regular if

1 p↑ ∈ S for all p ∈ P, 2

S exists in P for all S ∈ S , and

3 whenever T ∈ P* \ S , there is T ′ ∈ P* with T ⊆ T ′,

p < T = ⇒ p < T ′ for all p ∈ P, and for all S ∈ S , S ⊆ T ′ = ⇒ S ∈ T ′.

Definition (S -closure)

If S ⊆ P* we say a standard closure operator Γ: P*δ → P*δ is an S -closure if for all S ∈ P*, Γ(S) = p↑ for some p ∈ P ⇐ ⇒ S ∈ S .

Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 9 / 23

What meets can be preserved? part III

Proposition

Let P be a poset, and let S ⊆ P*. Then S is regular if and only if there exists an S -closure.

Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 10 / 23

What meets can be preserved? part III

Proposition

Let P be a poset, and let S ⊆ P*. Then S is regular if and only if there exists an S -closure.

Proof.

That S is regular whenever there exists an S -closure is a straightforward checking of the definition. We shall see that when S -is regular there exists an S -closure by direct construction.

Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 10 / 23

Constructing an S closure

Definition (FS , ΓS )

If S ⊆ P* is regular define FS = {f ∈ P* : S ⊆ f = ⇒ S ∈ f for all S ∈ S }, and define ΓS : P*δ → P*δ by ΓS (S) = {f ∈ FS : S ⊆ f }.

Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 11 / 23

Constructing an S closure

Definition (FS , ΓS )

If S ⊆ P* is regular define FS = {f ∈ P* : S ⊆ f = ⇒ S ∈ f for all S ∈ S }, and define ΓS : P*δ → P*δ by ΓS (S) = {f ∈ FS : S ⊆ f }.

Lemma

If S ⊆ P* is regular then ΓS is an S -closure.

Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 11 / 23

Constructing an S closure

Definition (FS , ΓS )

If S ⊆ P* is regular define FS = {f ∈ P* : S ⊆ f = ⇒ S ∈ f for all S ∈ S }, and define ΓS : P*δ → P*δ by ΓS (S) = {f ∈ FS : S ⊆ f }.

Lemma

If S ⊆ P* is regular then ΓS is an S -closure.

Proof.

It’s easy to see that ΓS is a standard closure operator on P*δ, and that ΓS (S) = p↑ for some p ∈ P whenever S ∈ S . If T ∈ P* \ S then there is T ′ ∈ FS with T ⊆ T ′ ⊂ p↑ whenever p is a lower bound for T, and since we must have ΓS (T) ⊆ ΓS (T ′) = T ′ ⊂ p↑ for all lower bounds p of T we cannot have ΓS (T) = p↑ for any p ∈ P, as ΓS (T) = p↑ = ⇒ p < T.

Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 11 / 23

How does this answer our question?

Lemma

et e : P → Q be an embedding, let S ∈ P*, and suppose S = p in P. Then e[S] = e(p) in Q if and only if Γe(S) = p↑.

Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 12 / 23

How does this answer our question?

Lemma

et e : P → Q be an embedding, let S ∈ P*, and suppose S = p in P. Then e[S] = e(p) in Q if and only if Γe(S) = p↑.

Proposition

Let P be a poset and let S ⊆ P*, then S is regular if and only if there is an S -regular meet-completion of P.

Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 12 / 23

How does this answer our question?

Lemma

et e : P → Q be an embedding, let S ∈ P*, and suppose S = p in P. Then e[S] = e(p) in Q if and only if Γe(S) = p↑.

Proposition

Let P be a poset and let S ⊆ P*, then S is regular if and only if there is an S -regular meet-completion of P.

Proof.

If S is regular just take ΓS and the lemma above gives the result. Conversely, a meet-completion e : P → Q gives rise to a standard closure

• perator Γe, which by the lemma must be an S -closure, which means S

must be regular.

Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 12 / 23

How does this answer our question? part II

Theorem

If S is regular then then the set MS ⊆ MP of S -regular meet-completions is dually order isomorphic to the set SS ⊆ SP of S -closures.

Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 13 / 23

The structure of SS

Proposition

SS is a lattice, and is closed under arbitrary non-empty meets.

Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 14 / 23

The structure of SS

Proposition

SS is a lattice, and is closed under arbitrary non-empty meets.

Proof.

This follows easily from the fact that the intersection of closed sets is closed, and that given S -closures Γ1 and Γ2 the composition Γ2 ◦ Γ1 is also an S -closure, and is an upper bound for {Γ1, Γ2} in SS .

Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 14 / 23

The structure of SS

Proposition

SS is a lattice, and is closed under arbitrary non-empty meets.

Proof.

This follows easily from the fact that the intersection of closed sets is closed, and that given S -closures Γ1 and Γ2 the composition Γ2 ◦ Γ1 is also an S -closure, and is an upper bound for {Γ1, Γ2} in SS .

Lemma

ΓS is the bottom element of SS .

Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 14 / 23

The structure of SS

Proposition

SS is a lattice, and is closed under arbitrary non-empty meets.

Proof.

This follows easily from the fact that the intersection of closed sets is closed, and that given S -closures Γ1 and Γ2 the composition Γ2 ◦ Γ1 is also an S -closure, and is an upper bound for {Γ1, Γ2} in SS .

Lemma

ΓS is the bottom element of SS .

Proof.

Let Γ be an S -closure, let T ∈ P*δ, and let S ∈ S , then S ⊆ Γ(T) = ⇒ Γ(S) ⊆ Γ(T) ⇐ ⇒ S ∈ Γ(T), so Γ(T) ∈ FS and thus ΓS (T) ⊆ Γ(T) and we have the result.

Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 14 / 23

Does SS have a top element?

Let P be a poset and let S ⊆ P* be meet-closed.

Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 15 / 23

Does SS have a top element?

Let P be a poset and let S ⊆ P* be meet-closed. For each T ∈ P* \ S define ¯ T = {T ′ ∈ FS : T ⊆ T ′, and p < T = ⇒ p < T ′ for all p ∈ P}.

Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 15 / 23

Does SS have a top element?

Let P be a poset and let S ⊆ P* be meet-closed. For each T ∈ P* \ S define ¯ T = {T ′ ∈ FS : T ⊆ T ′, and p < T = ⇒ p < T ′ for all p ∈ P}.

Proposition

SS has a top element if and only if for each T ∈ P* \ S there is T m ∈ ¯ T such that for every Y ⊆ {¯ U : U ∈ P* \ S } with Y ∩ ¯ U = ∅ for all U ∈ P* \ S we have {f ∈ Y : T ⊆ f } ⊆ T m.

Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 15 / 23

Does SS have a bottom element? part II

Proof sketch:

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

Does SS have a bottom element? part II

Proof sketch: Suppose such T m exist and wlog that each is minimal (possible as ¯ T is closed under intersection).

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

Does SS have a bottom element? part II

Proof sketch: Suppose such T m exist and wlog that each is minimal (possible as ¯ T is closed under intersection). Define F′ = {p↑ : p ∈ P} ∪ {T m : T ∈ P* \ S }

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

Does SS have a bottom element? part II

Proof sketch: Suppose such T m exist and wlog that each is minimal (possible as ¯ T is closed under intersection). Define F′ = {p↑ : p ∈ P} ∪ {T m : T ∈ P* \ S } Define F to be the closure of F′ under arbitrary intersections

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

Does SS have a bottom element? part II

Proof sketch: Suppose such T m exist and wlog that each is minimal (possible as ¯ T is closed under intersection). Define F′ = {p↑ : p ∈ P} ∪ {T m : T ∈ P* \ S } Define F to be the closure of F′ under arbitrary intersections Define Γ: U → {f ∈ F : U ⊆ f }

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

Does SS have a bottom element? part II

Proof sketch: Suppose such T m exist and wlog that each is minimal (possible as ¯ T is closed under intersection). Define F′ = {p↑ : p ∈ P} ∪ {T m : T ∈ P* \ S } Define F to be the closure of F′ under arbitrary intersections Define Γ: U → {f ∈ F : U ⊆ f } And for the converse:

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

Does SS have a bottom element? part II

Proof sketch: Suppose such T m exist and wlog that each is minimal (possible as ¯ T is closed under intersection). Define F′ = {p↑ : p ∈ P} ∪ {T m : T ∈ P* \ S } Define F to be the closure of F′ under arbitrary intersections Define Γ: U → {f ∈ F : U ⊆ f } And for the converse: Suppose Γ′ is the top element for SS

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

Does SS have a bottom element? part II

Proof sketch: Suppose such T m exist and wlog that each is minimal (possible as ¯ T is closed under intersection). Define F′ = {p↑ : p ∈ P} ∪ {T m : T ∈ P* \ S } Define F to be the closure of F′ under arbitrary intersections Define Γ: U → {f ∈ F : U ⊆ f } And for the converse: Suppose Γ′ is the top element for SS Define T m = Γ′(T)

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

Does SS have a top element? part III

Corollary

If ¯ T has at least one maximal element (ordering by inclusion) for all T ∈ P* \ S then SS has a top element and thus is complete.

Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 17 / 23

Does SS have a top element? part III

Corollary

If ¯ T has at least one maximal element (ordering by inclusion) for all T ∈ P* \ S then SS has a top element and thus is complete.

Proof.

Given T ∈ P* \ S define MT ⊆ ¯ T to be the set of maximal elements of ¯ T, and define T m = MT.

Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 17 / 23

SS without a top element

• a
• b
• Robert Egrot, with thanks to Robin Hirsch ()

Preserving meets in meet-completions February 16, 2012 18 / 23

SS without a top element

• a
• b
• Define S = {p↑ : p ∈ P} ∪ {S ∈ P* : |S| = |ω|}

Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 18 / 23

SS without a top element

• a
• b
• Define S = {p↑ : p ∈ P} ∪ {S ∈ P* : |S| = |ω|}

Let U = {a, b}

Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 18 / 23

SS without a top element

• a
• b
• Define S = {p↑ : p ∈ P} ∪ {S ∈ P* : |S| = |ω|}

Let U = {a, b} If U′ ∈ ¯ U then U′ must be finite, so there is p ∈ P \ (U′ ∪ {0})

Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 18 / 23

SS without a top element

• a
• b
• Define S = {p↑ : p ∈ P} ∪ {S ∈ P* : |S| = |ω|}

Let U = {a, b} If U′ ∈ ¯ U then U′ must be finite, so there is p ∈ P \ (U′ ∪ {0}) Define Y = {f ∈ P* : f is finite and U′ ∪ {p} ⊆ f }, then

Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 18 / 23

SS without a top element

• a
• b
• Define S = {p↑ : p ∈ P} ∪ {S ∈ P* : |S| = |ω|}

Let U = {a, b} If U′ ∈ ¯ U then U′ must be finite, so there is p ∈ P \ (U′ ∪ {0}) Define Y = {f ∈ P* : f is finite and U′ ∪ {p} ⊆ f }, then

Y ⊆ { ¯ T : T ∈ P* \ S }

Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 18 / 23

SS without a top element

• a
• b
• Define S = {p↑ : p ∈ P} ∪ {S ∈ P* : |S| = |ω|}

Let U = {a, b} If U′ ∈ ¯ U then U′ must be finite, so there is p ∈ P \ (U′ ∪ {0}) Define Y = {f ∈ P* : f is finite and U′ ∪ {p} ⊆ f }, then

Y ⊆ { ¯ T : T ∈ P* \ S } Y ∩ ¯ T = ∅ for all T ∈ P* \ S

Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 18 / 23

SS without a top element

• a
• b
• Define S = {p↑ : p ∈ P} ∪ {S ∈ P* : |S| = |ω|}

Let U = {a, b} If U′ ∈ ¯ U then U′ must be finite, so there is p ∈ P \ (U′ ∪ {0}) Define Y = {f ∈ P* : f is finite and U′ ∪ {p} ⊆ f }, then

Y ⊆ { ¯ T : T ∈ P* \ S } Y ∩ ¯ T = ∅ for all T ∈ P* \ S {f ∈ Y : U ⊆ s} ⊇ U′ ∪ {p} ⊃ U′

Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 18 / 23

SS without a top element

• a
• b
• Define S = {p↑ : p ∈ P} ∪ {S ∈ P* : |S| = |ω|}

Let U = {a, b} If U′ ∈ ¯ U then U′ must be finite, so there is p ∈ P \ (U′ ∪ {0}) Define Y = {f ∈ P* : f is finite and U′ ∪ {p} ⊆ f }, then

Y ⊆ { ¯ T : T ∈ P* \ S } Y ∩ ¯ T = ∅ for all T ∈ P* \ S {f ∈ Y : U ⊆ s} ⊇ U′ ∪ {p} ⊃ U′

So, SS has no top element.

Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 18 / 23

The structure of SS , part I

If L is a complete lattice let CL(L) be the complete lattice of all standard closure operators on L (ordered pointwise).

Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 19 / 23

The structure of SS , part I

If L is a complete lattice let CL(L) be the complete lattice of all standard closure operators on L (ordered pointwise).

Lemma

If S ⊆ CL(L) is closed under finite joins (inherited from CL(L)), and whenever Γ1 ≤ Γ2 ∈ S the interval [Γ1, Γ2] ⊆ S, then S is weakly upper semimodular as a poset.

Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 19 / 23

The structure of SS , part I

If L is a complete lattice let CL(L) be the complete lattice of all standard closure operators on L (ordered pointwise).

Lemma

If S ⊆ CL(L) is closed under finite joins (inherited from CL(L)), and whenever Γ1 ≤ Γ2 ∈ S the interval [Γ1, Γ2] ⊆ S, then S is weakly upper semimodular as a poset.

Proof.

This is a slight generalization of [1, lemma 3], which in turn calls upon some general theory from [2].

Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 19 / 23

The structure of SS , part I

If L is a complete lattice let CL(L) be the complete lattice of all standard closure operators on L (ordered pointwise).

Lemma

If S ⊆ CL(L) is closed under finite joins (inherited from CL(L)), and whenever Γ1 ≤ Γ2 ∈ S the interval [Γ1, Γ2] ⊆ S, then S is weakly upper semimodular as a poset.

Proof.

This is a slight generalization of [1, lemma 3], which in turn calls upon some general theory from [2].

Corollary

When S is regular SS is weakly upper semimodular.

Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 19 / 23

The structure of SS , part II

Theorem

[1, theorem 6] If S is a finite subinterval of CL(L) then then S is upper semimodular, an upper bounded homomorphic image of a free lattice, and thus meet semidistributive.

Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 20 / 23

The structure of SS , part II

Theorem

[1, theorem 6] If S is a finite subinterval of CL(L) then then S is upper semimodular, an upper bounded homomorphic image of a free lattice, and thus meet semidistributive.

Corollary

If P is finite, and if S ⊆ P* is regular, then SS is upper semimodular, an upper bounded homomorphic image of a free lattice, and thus meet semidistributive.

Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 20 / 23

The structure of SS , appendix part I

Definition (weakly lower/upper semimodular)

A poset P is weakly lower semimodular if whenever a, b, c ∈ P with a = b, a ≺ c and b ≺ c there is d with d ≺ a and d ≺ b. Weakly upper semimodularity is defined dually

Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 21 / 23

The structure of SS , appendix part I

Definition (weakly lower/upper semimodular)

A poset P is weakly lower semimodular if whenever a, b, c ∈ P with a = b, a ≺ c and b ≺ c there is d with d ≺ a and d ≺ b. Weakly upper semimodularity is defined dually

Definition (lower/upper semimodular)

A lattice L is lower semimodular if for all a, b, c ∈ L, if a b and a < b < a ∨ c then there is some d ∈ L with c ≤ d < a ∨ c and a ∨ (b ∧ d) = b. Upper semimodularity is defined dually.

Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 21 / 23

The structure of SS , appendix part I

Definition (weakly lower/upper semimodular)

A poset P is weakly lower semimodular if whenever a, b, c ∈ P with a = b, a ≺ c and b ≺ c there is d with d ≺ a and d ≺ b. Weakly upper semimodularity is defined dually

Definition (lower/upper semimodular)

A lattice L is lower semimodular if for all a, b, c ∈ L, if a b and a < b < a ∨ c then there is some d ∈ L with c ≤ d < a ∨ c and a ∨ (b ∧ d) = b. Upper semimodularity is defined dually. For lattices weak lower/upper semimodularity are implied by lower/upper semimodularity respectively, and in a dually strongly atomic lattice the converse also holds (a lattice is strongly atomic if whenever a < b there is p with a ≺ p ≤ b).

Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 21 / 23

The structure of SS , appendix part II

Definition (upper/lower bounded homomorphism)

A lattice homomorphism h: K → L is upper bounded if {b ∈ K : h(b) ≤ a} is either empty or has a greatest element for all a ∈ L. Lower bounded homomorphisms are defined dually.

Robert Egrot, with thanks to Robin Hirsch () Preserving meets in meet-completions February 16, 2012 22 / 23

Bibliography

Nation, J.B., Pogel, A.: The lattice of completions of an ordered set. Order 14, 1–7 (1997) Ward, M.: The closure operators of a lattice.

• Ann. Math. 43, 191–196 (1942)

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