Completions of semilattices

joint work with Hilary Priestley

TACL July 27, 2011

Maria Jo˜aoGouveia Universidade de Lisboa Finitely generated varieties of lattice-based algebras

Profinite completions

Canonical extensions

Natural extensions ˆ I the inverse limit A of the family

{ A/θ | θ ∈ Conf A },

where Conf A is the set of congruences on A of finite index

I the natural homomorphisms A → A/θ separate the elements of A A embeds into Aˆ via the embedding

a 7→ (a/θ)θ∈Conf A

Profinite completion of an algebra A I the natural homomorphisms A → A/θ separate the elements of A A embeds into Aˆ via the embedding

a 7→ (a/θ)θ∈Conf A

Profinite completion of an algebra A

ˆ I the inverse limit A of the family

{ A/θ | θ ∈ Conf A },

where Conf A is the set of congruences on A of finite index Profinite completion of an algebra A

ˆ I the inverse limit A of the family

{ A/θ | θ ∈ Conf A },

where Conf A is the set of congruences on A of finite index

I the natural homomorphisms A → A/θ separate the elements of A A embeds into Aˆ via the embedding

a 7→ (a/θ)θ∈Conf A I a complete lattice into which A embeds via an embedding e I dense i.e. each of its elements is a join of meets and a meet of joins of elements of e(A) I and compact i.e. ^ _ e(F ) 6 e(I ) ⇔ F ∩ I 6= ∅ for every filter F and ideal I of A

Canonical extension of a bounded lattice A I dense i.e. each of its elements is a join of meets and a meet of joins of elements of e(A) I and compact i.e. ^ _ e(F ) 6 e(I ) ⇔ F ∩ I 6= ∅ for every filter F and ideal I of A

Canonical extension of a bounded lattice A

I a complete lattice into which A embeds via an embedding e i.e. each of its elements is a join of meets and a meet of joins of elements of e(A) I and compact i.e. ^ _ e(F ) 6 e(I ) ⇔ F ∩ I 6= ∅ for every filter F and ideal I of A

Canonical extension of a bounded lattice A

I a complete lattice into which A embeds via an embedding e I dense I and compact i.e. ^ _ e(F ) 6 e(I ) ⇔ F ∩ I 6= ∅ for every filter F and ideal I of A

Canonical extension of a bounded lattice A

I a complete lattice into which A embeds via an embedding e I dense i.e. each of its elements is a join of meets and a meet of joins of elements of e(A) i.e. ^ _ e(F ) 6 e(I ) ⇔ F ∩ I 6= ∅ for every filter F and ideal I of A

Canonical extension of a bounded lattice A

I a complete lattice into which A embeds via an embedding e I dense i.e. each of its elements is a join of meets and a meet of joins of elements of e(A) I and compact Canonical extension of a bounded lattice A

I a complete lattice into which A embeds via an embedding e I dense i.e. each of its elements is a join of meets and a meet of joins of elements of e(A) I and compact i.e. ^ _ e(F ) 6 e(I ) ⇔ F ∩ I 6= ∅ for every filter F and ideal I of A Aδ denotes the canonical extension of A

Harding 2006 δ ∼ ˆ I A = A

I a canonical extension of Gehrke and Harding A always exists 2001 I it is unique up to isomorphism Harding 2006 δ ∼ ˆ I A = A

I a canonical extension of Gehrke and Harding A always exists 2001 I it is unique up to isomorphism

Aδ denotes the canonical extension of A I a canonical extension of Gehrke and Harding A always exists 2001 I it is unique up to isomorphism

Aδ denotes the canonical extension of A

Harding 2006 δ ∼ ˆ I A = A Let A ∈ V = ISP(M), where M is finite

I the map e : A → MV(A,M) a 7→ e(a): V(A, M) → M h 7→ h(a)

is an embedding

I the natural extension nV (A) is the topological V(A,M) closure of e(A) in the topological space MT

Natural extension of an algebra A I the map e : A → MV(A,M) a 7→ e(a): V(A, M) → M h 7→ h(a)

is an embedding

I the natural extension nV (A) is the topological V(A,M) closure of e(A) in the topological space MT

Natural extension of an algebra A

Let A ∈ V = ISP(M), where M is finite I the natural extension nV (A) is the topological V(A,M) closure of e(A) in the topological space MT

Natural extension of an algebra A

Let A ∈ V = ISP(M), where M is finite

I the map e : A → MV(A,M) a 7→ e(a): V(A, M) → M h 7→ h(a)

is an embedding Natural extension of an algebra A

Let A ∈ V = ISP(M), where M is finite

I the map e : A → MV(A,M) a 7→ e(a): V(A, M) → M h 7→ h(a)

is an embedding

I the natural extension nV (A) is the topological V(A,M) closure of e(A) in the topological space MT If R ⊆ S (Mn) entails S (Mn), then I n∈N S n∈N S

nV (A) = {R-preserving maps V(A, M) → M}

and so

δ ∼ ∼ A = Aˆ = nV (A) = {R-preserving maps V(A, M) → M}

Davey, G, Haviar and Priestley (in press)

ˆ ∼ I A = nV (A) and so

δ ∼ ∼ A = Aˆ = nV (A) = {R-preserving maps V(A, M) → M}

Davey, G, Haviar and Priestley (in press)

ˆ ∼ I A = nV (A)

If R ⊆ S (Mn) entails S (Mn), then I n∈N S n∈N S

nV (A) = {R-preserving maps V(A, M) → M} Davey, G, Haviar and Priestley (in press)

ˆ ∼ I A = nV (A)

If R ⊆ S (Mn) entails S (Mn), then I n∈N S n∈N S

nV (A) = {R-preserving maps V(A, M) → M}

and so

δ ∼ ∼ A = Aˆ = nV (A) = {R-preserving maps V(A, M) → M} I S = ISP(M), where M = h{0, 1}; ∨, 0i

Hofmann, Mislove, Stralka {∨, 0} entails S (Mn) I n∈N S 1974

. Aˆ is formed by all the {∨, 0}-preserving maps S(A, M) to M

S - the variety of join-semilattices with 0 Hofmann, Mislove, Stralka {∨, 0} entails S (Mn) I n∈N S 1974

. Aˆ is formed by all the {∨, 0}-preserving maps S(A, M) to M

S - the variety of join-semilattices with 0

I S = ISP(M), where M = h{0, 1}; ∨, 0i . Aˆ is formed by all the {∨, 0}-preserving maps S(A, M) to M

S - the variety of join-semilattices with 0

I S = ISP(M), where M = h{0, 1}; ∨, 0i

Hofmann, Mislove, Stralka {∨, 0} entails S (Mn) I n∈N S 1974 S - the variety of join-semilattices with 0

I S = ISP(M), where M = h{0, 1}; ∨, 0i

Hofmann, Mislove, Stralka {∨, 0} entails S (Mn) I n∈N S 1974

. Aˆ is formed by all the {∨, 0}-preserving maps S(A, M) to M S - the variety of join-semilattices with 0

I S = ISP(M), where M = h{0, 1}; ∨, 0i

Hofmann, Mislove, Stralka {∨, 0} entails S (Mn) I n∈N S 1974

ˆ I A is the join-semilattice S(S(A, M), M) I Existence and uniqueness of a canonical extension of a poset (Dunn, Gehrke, Palmigiano 2005)

I The canonical extension of A is a compact and dense completion C relative to some order embedding e : A ,→ C

Let A be a semilattice in S

I A is a poset I The canonical extension of A is a compact and dense completion C relative to some order embedding e : A ,→ C

Let A be a semilattice in S

I A is a poset

I Existence and uniqueness of a canonical extension of a poset (Dunn, Gehrke, Palmigiano 2005) Let A be a semilattice in S

I A is a poset

I Existence and uniqueness of a canonical extension of a poset (Dunn, Gehrke, Palmigiano 2005)

I The canonical extension of A is a compact and dense completion C relative to some order embedding e : A ,→ C C is a completion of A if C is a complete lattice and there exists an order embedding e : A ,→ C C is compact if ^ _ e(F ) 6 e(I ) ⇔ F ∩ I 6= ∅ for every down-directed up-set F and every up-directed down-set I of A C is dense if every element of C is a join of meets of down-directed up-sets of e(A) and a meet of joins of up-directed down-sets of e(A) I Can we relate the canonical extension of A and its profinite completion Aˆ?

Let A be a semilattice in S

I A is a poset

I Existence and uniqueness of a canonical extension of a poset (Dunn, Gehrke, Palmigiano 2005)

I The canonical extension of A is a compact and dense completion C relative to some order embedding e : A ,→ C Let A be a semilattice in S

I A is a poset

I Existence and uniqueness of a canonical extension of a poset (Dunn, Gehrke, Palmigiano 2005)

I The canonical extension of A is a compact and dense completion C relative to some order embedding e : A ,→ C

I Can we relate the canonical extension of A and its profinite completion Aˆ? • @ @ • · • · · · · cn+1 · · · @ an+1 bn+1 bc@n @ an b ·b @b bn · ·b · ·b · • · @ • @• @ @ N

0 b

Ab

Example 1

· · · · · cn+1 · · · @ an+1 bn+1 bc@n @ an b ·b @b bn · ·b · ·b · · 0 b A Example 1

• @ • · @• · · · c · · · n+1 · · · · · · · cn+1 · @ · · an+1 bn+1 · · bc@n @ an+1 b @ bn+1 @ cn an b ·b @b bn @ · an b b @b bn · · · · ·b ·b · · • · ·b · ·b · · · · @ • @• @ 0 @ b N

A 0 b

Ab ˆ I A is a complete lattice ˆ I Every element of A is a meet of joins of elements of e(A)

ˆ I A is compact

Properties of Aˆ ˆ I Every element of A is a meet of joins of elements of e(A)

ˆ I A is compact

Properties of Aˆ

ˆ I A is a complete lattice ˆ I A is compact

Properties of Aˆ

ˆ I A is a complete lattice ˆ I Every element of A is a meet of joins of elements of e(A) Properties of Aˆ

ˆ I A is a complete lattice ˆ I Every element of A is a meet of joins of elements of e(A)

ˆ I A is compact ˆ I C∨(A) is a subsemilattice of A

I C∨(A) is a complete lattice

ˆ I Directed meets coincide in C∨(A) and A and so C∨(A) is compact

I C∨(A) is dense

Define C∨(A) := { joins of directed meets of e(A)}

I C∨(A) contains e(A), 0Aˆ and 1Aˆ I C∨(A) is a complete lattice

ˆ I Directed meets coincide in C∨(A) and A and so C∨(A) is compact

I C∨(A) is dense

Define C∨(A) := { joins of directed meets of e(A)}

I C∨(A) contains e(A), 0Aˆ and 1Aˆ

ˆ I C∨(A) is a subsemilattice of A ˆ I Directed meets coincide in C∨(A) and A and so C∨(A) is compact

I C∨(A) is dense

Define C∨(A) := { joins of directed meets of e(A)}

I C∨(A) contains e(A), 0Aˆ and 1Aˆ

ˆ I C∨(A) is a subsemilattice of A

I C∨(A) is a complete lattice and so C∨(A) is compact

I C∨(A) is dense

Define C∨(A) := { joins of directed meets of e(A)}

I C∨(A) contains e(A), 0Aˆ and 1Aˆ

ˆ I C∨(A) is a subsemilattice of A

I C∨(A) is a complete lattice

ˆ I Directed meets coincide in C∨(A) and A I C∨(A) is dense

Define C∨(A) := { joins of directed meets of e(A)}

I C∨(A) contains e(A), 0Aˆ and 1Aˆ

ˆ I C∨(A) is a subsemilattice of A

I C∨(A) is a complete lattice

ˆ I Directed meets coincide in C∨(A) and A and so C∨(A) is compact Define C∨(A) := { joins of directed meets of e(A)}

I C∨(A) contains e(A), 0Aˆ and 1Aˆ

ˆ I C∨(A) is a subsemilattice of A

I C∨(A) is a complete lattice

ˆ I Directed meets coincide in C∨(A) and A and so C∨(A) is compact

I C∨(A) is dense We denote the canonical extension of a δ semilattice A ∈ S by A∨.

Theorem The canonical extension of a semilattice A ∈ S is the subsemilattice of Ab formed by all joins of meets of down-directed up-sets of e(A). Theorem The canonical extension of a semilattice A ∈ S is the subsemilattice of Ab formed by all joins of meets of down-directed up-sets of e(A).

We denote the canonical extension of a δ semilattice A ∈ S by A∨. Proposition δ Let A be a semilattice in S. Then A∨ satisfies the WV restricted distributive law:

_ ^ { e(Y ) | Y ∈ Y } = ^ _ { e(Z) | Z ⊆ A, ∀Y ∈ Y Z ∩ Y 6= ∅ }, for every family Y of down-directed subsets of A. Theorem

Let L be a lattice with 0 and let A = L∨ be its semilattice reduct in S δ The lattice A∨ is the canonical extension of L. x0 •b

x1 • · x2 · • · · x ·· · N · · · b

··· b b

b b b ··· H H@ b bHH@b b Ab

A = L∨

Example 2

b · · · · · · b

··· b b

b b b ··· H H@ b bHH@b b L Example 2

x0 •b

x1 b • · · · x2 · · • · · · x ·· · · N · · · · b · b

··· b b ··· b b

b b b b b b ··· H ··· H H@ H@ b bHH@b b bHH@b b b L Ab

A = L∨ Proposition Let L be a bounded distributive lattice.

Suppose the poset IP (L) of prime ideals of L has finite width. The canonical extension of L is isomorphic to the profinite completion Lc∨ of its ∨-semilattice reduct L∨. Thank you!