On Lattice-Valued Frames: the Completely Distributive Case

On lattice-valued frames: the completely distributive case

Javier Gutiérrez García

(joint work with Ulrich Höhle and M. Angeles de Prada Vicente)

Chengdu, April 6, 2010

On lattice-valued frames Dedicated to Professor Liu on his 70th birthday

On lattice-valued frames On lattice-valued frames On lattice-valued frames On lattice-valued frames Origin of the work

On lattice-valued frames • Visit of Ulrich Höhle to Bilbao, April 4 to 11, (2008)

• 29th Linz Seminar on Fuzzy Set Theory: “Foundations of Lattice-Valued Mathematics with Applications to Algebra and Topology” Linz, (Austria), February 4 to 9, (2008) Attendants: Rodabaugh, Zhang, Höhle, Kubiak, Šostak, de Prada Vicente,. . .

A. Pultr, S.E. Rodabaugh, Category theoretic aspects of chain-valued frames: Part I: Categorical and presheaf theoretic foundations, Part II: Applications to lattice-valued topology, Fuzzy Sets and Systems 159 (2008) 501–528 and 529–558.

On lattice-valued frames • Visit of Ulrich Höhle to Bilbao, April 4 to 11, (2008)

On lattice-valued frames • 29th Linz Seminar on Fuzzy Set Theory: “Foundations of Lattice-Valued Mathematics with Applications to Algebra and Topology” Linz, (Austria), February 4 to 9, (2008) Attendants: Rodabaugh, Zhang, Höhle, Kubiak, Šostak, de Prada Vicente,. . .

• Visit of Ulrich Höhle to Bilbao, April 4 to 11, (2008)

On lattice-valued frames • L-valued (Fuzzy) topology

• Pointfree (Pointless) topology, Frame (Locale) theory

Introduction

On lattice-valued frames • L-valued (Fuzzy) topology

Introduction

• Pointfree (Pointless) topology, Frame (Locale) theory

On lattice-valued frames Introduction

• Pointfree (Pointless) topology, Frame (Locale) theory

• L-valued (Fuzzy) topology

On lattice-valued frames Pointfree topology

On lattice-valued frames POINTFREE TOPOLOGY

S S A ∩ Bi = (A ∩ Bi ) i∈I i∈I

f −1 preserves S and ∩

(Y , OY ) (OY , ⊆)

Abstraction TOPOLOGY /o /o /o /o /o /o /o /o /o /o /o /

Pointfree topology Motivation

(X, OX) /o /o /o /o /o /o /o / (OX, ⊆)

On lattice-valued frames POINTFREE TOPOLOGY

f −1 preserves S and ∩

(Y , OY ) (OY , ⊆)

Abstraction TOPOLOGY /o /o /o /o /o /o /o /o /o /o /o /

Pointfree topology Motivation

S S (X, OX) /o /o /o /o /o /o /o / (OX, ⊆) A ∩ Bi = (A ∩ Bi ) i∈I i∈I

On lattice-valued frames POINTFREE TOPOLOGY

f −1 preserves S and ∩

(OY , ⊆)

Abstraction TOPOLOGY /o /o /o /o /o /o /o /o /o /o /o /

Pointfree topology Motivation

S S (X, OX) /o /o /o /o /o /o /o / (OX, ⊆) A ∩ Bi = (A ∩ Bi ) i∈I i∈I

(Y , OY )

On lattice-valued frames POINTFREE TOPOLOGY

f −1 preserves S and ∩

Abstraction TOPOLOGY /o /o /o /o /o /o /o /o /o /o /o /

Pointfree topology Motivation

S S (X, OX) /o /o /o /o /o /o /o / (OX, ⊆) A ∩ Bi = (A ∩ Bi ) i∈I i∈I

(Y , OY ) /o /o /o /o /o /o /o / (OY , ⊆)

On lattice-valued frames POINTFREE TOPOLOGY

f −1 preserves S and ∩

Abstraction TOPOLOGY /o /o /o /o /o /o /o /o /o /o /o /

Pointfree topology Motivation

S S (X, OX) /o /o /o /o /o /o /o / (OX, ⊆) A ∩ Bi = (A ∩ Bi ) O i∈I i∈I

f f −1

(Y , OY ) /o /o /o /o /o /o /o / (OY , ⊆)

On lattice-valued frames POINTFREE TOPOLOGY

Abstraction TOPOLOGY /o /o /o /o /o /o /o /o /o /o /o /

Pointfree topology Motivation

S S (X, OX) /o /o /o /o /o /o /o / (OX, ⊆) A ∩ Bi = (A ∩ Bi ) O i∈I i∈I

f f −1 f −1 preserves S and ∩

(Y , OY ) /o /o /o /o /o /o /o / (OY , ⊆)

On lattice-valued frames POINTFREE TOPOLOGY

Pointfree topology Motivation

S S (X, OX) /o /o /o /o /o /o /o / (OX, ⊆) A ∩ Bi = (A ∩ Bi ) O i∈I i∈I

f f −1 f −1 preserves S and ∩

(Y , OY ) /o /o /o /o /o /o /o / (OY , ⊆)

Abstraction TOPOLOGY /o /o /o /o /o /o /o /o /o /o /o /

On lattice-valued frames Pointfree topology Motivation

S S (X, OX) /o /o /o /o /o /o /o / (OX, ⊆) A ∩ Bi = (A ∩ Bi ) O i∈I i∈I

f f −1 f −1 preserves S and ∩

(Y , OY ) /o /o /o /o /o /o /o / (OY , ⊆)

Abstraction TOPOLOGY /o /o /o /o /o /o /o /o /o /o /o / POINTFREE TOPOLOGY

On lattice-valued frames in which

∗ complete lattices L W W ∗ a ∧ i∈I ai = {a ∧ ai : i ∈ I} for all a ∈ L and {ai : i ∈ I} ⊆ L.

• Morphisms, called frame homomorphisms, are those maps between frames that preserve arbitrary joins and ﬁnite meets.

• O : Top → Frm is a contravariant functor with X 7−→ OX and f f −1 X → Y 7−→ OY −→ OX.

Pointfree topology the category of frames Frm

• The objects in Frm are frames, i.e.

On lattice-valued frames in which W W ∗ a ∧ i∈I ai = {a ∧ ai : i ∈ I} for all a ∈ L and {ai : i ∈ I} ⊆ L.

• Morphisms, called frame homomorphisms, are those maps between frames that preserve arbitrary joins and ﬁnite meets.

• O : Top → Frm is a contravariant functor with X 7−→ OX and f f −1 X → Y 7−→ OY −→ OX.

Pointfree topology the category of frames Frm

• The objects in Frm are frames, i.e.

∗ complete lattices L

On lattice-valued frames • Morphisms, called frame homomorphisms, are those maps between frames that preserve arbitrary joins and ﬁnite meets.

• O : Top → Frm is a contravariant functor with X 7−→ OX and f f −1 X → Y 7−→ OY −→ OX.

Pointfree topology the category of frames Frm

• The objects in Frm are frames, i.e.

∗ complete lattices L in which W W ∗ a ∧ i∈I ai = {a ∧ ai : i ∈ I} for all a ∈ L and {ai : i ∈ I} ⊆ L.

On lattice-valued frames • O : Top → Frm is a contravariant functor with X 7−→ OX and f f −1 X → Y 7−→ OY −→ OX.

Pointfree topology the category of frames Frm

• The objects in Frm are frames, i.e.

∗ complete lattices L in which W W ∗ a ∧ i∈I ai = {a ∧ ai : i ∈ I} for all a ∈ L and {ai : i ∈ I} ⊆ L.

• Morphisms, called frame homomorphisms, are those maps between frames that preserve arbitrary joins and ﬁnite meets.

On lattice-valued frames Pointfree topology the category of frames Frm

• The objects in Frm are frames, i.e.

∗ complete lattices L in which W W ∗ a ∧ i∈I ai = {a ∧ ai : i ∈ I} for all a ∈ L and {ai : i ∈ I} ⊆ L.

• Morphisms, called frame homomorphisms, are those maps between frames that preserve arbitrary joins and ﬁnite meets.

• O : Top → Frm is a contravariant functor with X 7−→ OX and f f −1 X → Y 7−→ OY −→ OX.

On lattice-valued frames • Morphisms, called localic maps, are of course, just frame homomorphisms taken backwards.

• O : Top → Loc is now a covariant functor with X 7−→ OX and f f −1 X → Y 7−→ OX −→ OY .

Advantage: Loc can be thought of as a natural extension of (sober) spaces.

Disadvantage: Morphisms thought in this way may obscure the intuition.

Pointfree topology the dual category Loc=Frmop

• The objects in Loc are frames, from now on, also called locales.

On lattice-valued frames • O : Top → Loc is now a covariant functor with X 7−→ OX and f f −1 X → Y 7−→ OX −→ OY .

Advantage: Loc can be thought of as a natural extension of (sober) spaces.

Disadvantage: Morphisms thought in this way may obscure the intuition.

Pointfree topology the dual category Loc=Frmop

• The objects in Loc are frames, from now on, also called locales.

• Morphisms, called localic maps, are of course, just frame homomorphisms taken backwards.

On lattice-valued frames Advantage: Loc can be thought of as a natural extension of (sober) spaces.

Disadvantage: Morphisms thought in this way may obscure the intuition.

Pointfree topology the dual category Loc=Frmop

• The objects in Loc are frames, from now on, also called locales.

• Morphisms, called localic maps, are of course, just frame homomorphisms taken backwards.

• O : Top → Loc is now a covariant functor with X 7−→ OX and f f −1 X → Y 7−→ OX −→ OY .

On lattice-valued frames Disadvantage: Morphisms thought in this way may obscure the intuition.

Pointfree topology the dual category Loc=Frmop

• The objects in Loc are frames, from now on, also called locales.

• Morphisms, called localic maps, are of course, just frame homomorphisms taken backwards.

• O : Top → Loc is now a covariant functor with X 7−→ OX and f f −1 X → Y 7−→ OX −→ OY .

Advantage: Loc can be thought of as a natural extension of (sober) spaces.

On lattice-valued frames Pointfree topology the dual category Loc=Frmop

• The objects in Loc are frames, from now on, also called locales.

• Morphisms, called localic maps, are of course, just frame homomorphisms taken backwards.

• O : Top → Loc is now a covariant functor with X 7−→ OX and f f −1 X → Y 7−→ OX −→ OY .

Advantage: Loc can be thought of as a natural extension of (sober) spaces.

Disadvantage: Morphisms thought in this way may obscure the intuition.

On lattice-valued frames D. Zhang, Y. Liu, L-fuzzy version of Stone’s representation theorem for distributive lattices, Fuzzy Sets and Systems 76 (1995) 259-270.

“Locales not only “capture” or “model” the lattice theoretical behaviour of topological spaces, more importantly when we work in a universe where choice principles are not allowed, it is locales, not spaces, which provide the right context in which to do topology.” “As an illustration, we look at the topological and the localic versions of Stone’s representation theorem for distributive lattices. ... Hence, in absence of Prime Ideal Theorem, we can prove Stone’s representation theorem in the context of locales, but no longer in the context of spaces. ”

P.T. Johnstone, The point of pointless topology, Bull. Amer. Math. Soc. 8 (1983) 41-53.

Pointfree Topology, Pointless Topology, Frame Theory, Locale Theory. . .

On lattice-valued frames “As an illustration, we look at the topological and the localic versions of Stone’s representation theorem for distributive lattices. ... Hence, in absence of Prime Ideal Theorem, we can prove Stone’s representation theorem in the context of locales, but no longer in the context of spaces. ”

D. Zhang, Y. Liu, L-fuzzy version of Stone’s representation theorem for distributive lattices, Fuzzy Sets and Systems 76 (1995) 259-270.

Pointfree Topology, Pointless Topology, Frame Theory, Locale Theory. . . “Locales not only “capture” or “model” the lattice theoretical behaviour of topological spaces, more importantly when we work in a universe where choice principles are not allowed, it is locales, not spaces, which provide the right context in which to do topology.”

P.T. Johnstone, The point of pointless topology, Bull. Amer. Math. Soc. 8 (1983) 41-53.

On lattice-valued frames Pointfree Topology, Pointless Topology, Frame Theory, Locale Theory. . . “Locales not only “capture” or “model” the lattice theoretical behaviour of topological spaces, more importantly when we work in a universe where choice principles are not allowed, it is locales, not spaces, which provide the right context in which to do topology.” “As an illustration, we look at the topological and the localic versions of Stone’s representation theorem for distributive lattices. ... Hence, in absence of Prime Ideal Theorem, we can prove Stone’s representation theorem in the context of locales, but no longer in the context of spaces. ”

P.T. Johnstone, The point of pointless topology, Bull. Amer. Math. Soc. 8 (1983) 41-53.

D. Zhang, Y. Liu, L-fuzzy version of Stone’s representation theorem for distributive lattices, Fuzzy Sets and Systems 76 (1995) 259-270.

On lattice-valued frames An element p ∈ L \{1} is called prime if for each α, β ∈ L with

α ∧ β ≤ p =⇒ α ≤ p or β ≤ p.

We denote by Spec L the spectrum of L, i.e. the set of all prime elements of L.

The functor Spec assigns to each frame L its spectrum Spec L, endowed with the hull-kernel topology whose open sets are

∆L(α) = {p ∈ Spec L : α 6≤ p} = Spec L \ ↑α for α ∈ L.

Pointfree topology spatial frames and sober spaces

Apart from the functor O : Top → Frm, there is a functor in the opposite direction, the spectrum functor

Spec : Frm → Top

On lattice-valued frames The functor Spec assigns to each frame L its spectrum Spec L, endowed with the hull-kernel topology whose open sets are

∆L(α) = {p ∈ Spec L : α 6≤ p} = Spec L \ ↑α for α ∈ L.

Pointfree topology spatial frames and sober spaces

Apart from the functor O : Top → Frm, there is a functor in the opposite direction, the spectrum functor

Spec : Frm → Top

An element p ∈ L \{1} is called prime if for each α, β ∈ L with

α ∧ β ≤ p =⇒ α ≤ p or β ≤ p.

We denote by Spec L the spectrum of L, i.e. the set of all prime elements of L.

On lattice-valued frames Pointfree topology spatial frames and sober spaces

Apart from the functor O : Top → Frm, there is a functor in the opposite direction, the spectrum functor

Spec : Frm → Top

An element p ∈ L \{1} is called prime if for each α, β ∈ L with

α ∧ β ≤ p =⇒ α ≤ p or β ≤ p.

We denote by Spec L the spectrum of L, i.e. the set of all prime elements of L.

The functor Spec assigns to each frame L its spectrum Spec L, endowed with the hull-kernel topology whose open sets are

∆L(α) = {p ∈ Spec L : α 6≤ p} = Spec L \ ↑α for α ∈ L.

On lattice-valued frames Recall that a topological space X is sober if the only prime opens are those of the form X \ {x} for some x ∈ X and a frame L is spatial if L is generated by its prime elements, i.e. if

α = V{p ∈ Spec L : α ≤ p} = V (↑α ∩ Spec L) for all α ∈ L,

The categories Sob of sober topological spaces and SpatFrm of spatial frames are dual under the restrictions of the functors O and Spec. Sob ∼ SpatFrm

Pointfree topology spatial frames and sober spaces

We have an adjoint situation:

O / Top o Frm Spec

On lattice-valued frames The categories Sob of sober topological spaces and SpatFrm of spatial frames are dual under the restrictions of the functors O and Spec. Sob ∼ SpatFrm

Pointfree topology spatial frames and sober spaces

We have an adjoint situation:

O / Top o Frm Spec

Recall that a topological space X is sober if the only prime opens are those of the form X \ {x} for some x ∈ X and a frame L is spatial if L is generated by its prime elements, i.e. if

α = V{p ∈ Spec L : α ≤ p} = V (↑α ∩ Spec L) for all α ∈ L,

On lattice-valued frames Pointfree topology spatial frames and sober spaces

We have an adjoint situation:

O / Top o Frm Spec

Recall that a topological space X is sober if the only prime opens are those of the form X \ {x} for some x ∈ X and a frame L is spatial if L is generated by its prime elements, i.e. if

α = V{p ∈ Spec L : α ≤ p} = V (↑α ∩ Spec L) for all α ∈ L,

The categories Sob of sober topological spaces and SpatFrm of spatial frames are dual under the restrictions of the functors O and Spec. Sob ∼ SpatFrm

On lattice-valued frames L-valued topology

On lattice-valued frames If α ∈ L, the associated constant map is denoted α.

If f : X → Y and b ∈ LY we let f −1(b) = b ◦ f ∈ LX .

An L-valued topological space (shortly, an L-topological space) is a pair (X, τ) consisting of a set X and a subset τ of LX (the L-valued topology or L-topology on the set X) closed under ﬁnite meets and arbitrary joins.

Given two L-topological spaces (X, τ), (Y , σ) a map f : X → Y is an L-continuous map if the correspondence f −1(b) maps σ into τ. The resulting category will be denoted by L-Top.

L-valued topology the category L-Top

With L a complete lattice and X a set, LX is the complete lattice of all maps from X to L, called L-sets, in which

a ≤ b in LX iff a(x) ≤ b(x) for all x ∈ X.

On lattice-valued frames If f : X → Y and b ∈ LY we let f −1(b) = b ◦ f ∈ LX .

Given two L-topological spaces (X, τ), (Y , σ) a map f : X → Y is an L-continuous map if the correspondence f −1(b) maps σ into τ. The resulting category will be denoted by L-Top.

L-valued topology the category L-Top

With L a complete lattice and X a set, LX is the complete lattice of all maps from X to L, called L-sets, in which

a ≤ b in LX iff a(x) ≤ b(x) for all x ∈ X.

If α ∈ L, the associated constant map is denoted α.

On lattice-valued frames An L-valued topological space (shortly, an L-topological space) is a pair (X, τ) consisting of a set X and a subset τ of LX (the L-valued topology or L-topology on the set X) closed under ﬁnite meets and arbitrary joins.

Given two L-topological spaces (X, τ), (Y , σ) a map f : X → Y is an L-continuous map if the correspondence f −1(b) maps σ into τ. The resulting category will be denoted by L-Top.

L-valued topology the category L-Top

With L a complete lattice and X a set, LX is the complete lattice of all maps from X to L, called L-sets, in which

a ≤ b in LX iff a(x) ≤ b(x) for all x ∈ X.

If α ∈ L, the associated constant map is denoted α.

If f : X → Y and b ∈ LY we let f −1(b) = b ◦ f ∈ LX .

On lattice-valued frames Given two L-topological spaces (X, τ), (Y , σ) a map f : X → Y is an L-continuous map if the correspondence f −1(b) maps σ into τ. The resulting category will be denoted by L-Top.

L-valued topology the category L-Top

With L a complete lattice and X a set, LX is the complete lattice of all maps from X to L, called L-sets, in which

a ≤ b in LX iff a(x) ≤ b(x) for all x ∈ X.

If α ∈ L, the associated constant map is denoted α.

If f : X → Y and b ∈ LY we let f −1(b) = b ◦ f ∈ LX .

On lattice-valued frames L-valued topology the category L-Top

With L a complete lattice and X a set, LX is the complete lattice of all maps from X to L, called L-sets, in which

a ≤ b in LX iff a(x) ≤ b(x) for all x ∈ X.

If α ∈ L, the associated constant map is denoted α.

If f : X → Y and b ∈ LY we let f −1(b) = b ◦ f ∈ LX .

Given two L-topological spaces (X, τ), (Y , σ) a map f : X → Y is an L-continuous map if the correspondence f −1(b) maps σ into τ. The resulting category will be denoted by L-Top.

On lattice-valued frames “It is a natural and interesting question whether or not it is possible to establish a category to play the same role with respect to a given notion of fuzzy topology as that locales play for topological spaces.”

D. Zhang, Y. Liu, L-fuzzy version of Stone’s representation theorem for distributive lattices, Fuzzy Sets and Systems 76 (1995) 259-270.

On lattice-valued frames • Liu and Zang’s approach

• Rodabaugh’s approach

Previous approaches

On lattice-valued frames • Liu and Zang’s approach

Previous approaches

• Rodabaugh’s approach

On lattice-valued frames Previous approaches

• Rodabaugh’s approach

• Liu and Zang’s approach

On lattice-valued frames • If f :(X, τ) → (Y , σ) is an L-continuous map, then f −1 : σ → τ is a frame homomorphism.

If L is a frame then:

• Each L-topology on X, being a subframe of the frame LX , is a frame as well.

Hence we can construct the functor LO : L-Top → Frm

LO(X, τ) = τ LOf :(X, τ) → (Y , σ) = f −1 : σ → τ.

L-valued frames previous approaches (I)

Of course, in order to answer this question, from now on the complete lattice L will be additionally assume to be a frame.

On lattice-valued frames • If f :(X, τ) → (Y , σ) is an L-continuous map, then f −1 : σ → τ is a frame homomorphism.

Hence we can construct the functor LO : L-Top → Frm

LO(X, τ) = τ LOf :(X, τ) → (Y , σ) = f −1 : σ → τ.

L-valued frames previous approaches (I)

Of course, in order to answer this question, from now on the complete lattice L will be additionally assume to be a frame.

If L is a frame then:

• Each L-topology on X, being a subframe of the frame LX , is a frame as well.

On lattice-valued frames Hence we can construct the functor LO : L-Top → Frm

LO(X, τ) = τ LOf :(X, τ) → (Y , σ) = f −1 : σ → τ.

L-valued frames previous approaches (I)

Of course, in order to answer this question, from now on the complete lattice L will be additionally assume to be a frame.

If L is a frame then:

• Each L-topology on X, being a subframe of the frame LX , is a frame as well.

• If f :(X, τ) → (Y , σ) is an L-continuous map, then f −1 : σ → τ is a frame homomorphism.

On lattice-valued frames L-valued frames previous approaches (I)

Of course, in order to answer this question, from now on the complete lattice L will be additionally assume to be a frame.

If L is a frame then:

• Each L-topology on X, being a subframe of the frame LX , is a frame as well.

• If f :(X, τ) → (Y , σ) is an L-continuous map, then f −1 : σ → τ is a frame homomorphism.

Hence we can construct the functor LO : L-Top → Frm

LO(X, τ) = τ LOf :(X, τ) → (Y , σ) = f −1 : σ → τ.

On lattice-valued frames Rodabaugh’s approach

On lattice-valued frames This is deﬁned by introducing the notion of L-point of a frame A, which are deﬁned to be frame homomorphisms from A to L.

We have an adjoint situation:

OL / L-Top o Frm LSpec

with right unit ΨL and left unit ΦL.

L-valued frames previous approaches (I)

LO has a right adjoint called the L-spectrum functor LSpec.

S.E. Rodabaugh, Point set lattice theoretic topology, Fuzzy Sets and Systems 40 (1991) 296–345.

On lattice-valued frames We have an adjoint situation:

OL / L-Top o Frm LSpec

with right unit ΨL and left unit ΦL.

L-valued frames previous approaches (I)

LO has a right adjoint called the L-spectrum functor LSpec.

This is deﬁned by introducing the notion of L-point of a frame A, which are deﬁned to be frame homomorphisms from A to L.

S.E. Rodabaugh, Point set lattice theoretic topology, Fuzzy Sets and Systems 40 (1991) 296–345.

On lattice-valued frames L-valued frames previous approaches (I)

LO has a right adjoint called the L-spectrum functor LSpec.

This is deﬁned by introducing the notion of L-point of a frame A, which are deﬁned to be frame homomorphisms from A to L.

We have an adjoint situation:

OL / L-Top o Frm LSpec

with right unit ΨL and left unit ΦL. S.E. Rodabaugh, Point set lattice theoretic topology, Fuzzy Sets and Systems 40 (1991) 296–345.

On lattice-valued frames The categories Sob and L-SpatFrm are dual under the restrictions of the functors LO and LSpec.

L-Sob ∼ L-SpatFrm

The approach is not completely satisfactory. . . For example the fuzzy real line fails to be L-sober.

L-valued frames previous approaches (I)

Let us denote by L-SpatFrm the full subcategory of L-spatial frames (frames for which ΦL is injective) and L-Sob the full subcategory of L-sober spaces (L-topological spaces for which ΨL is bijective).

S.E. Rodabaugh, Point set lattice theoretic topology, Fuzzy Sets and Systems 40 (1991) 296–345.

On lattice-valued frames The approach is not completely satisfactory. . . For example the fuzzy real line fails to be L-sober.

L-valued frames previous approaches (I)

The categories Sob and L-SpatFrm are dual under the restrictions of the functors LO and LSpec.

L-Sob ∼ L-SpatFrm

S.E. Rodabaugh, Point set lattice theoretic topology, Fuzzy Sets and Systems 40 (1991) 296–345.

On lattice-valued frames L-valued frames previous approaches (I)

The categories Sob and L-SpatFrm are dual under the restrictions of the functors LO and LSpec.

L-Sob ∼ L-SpatFrm

The approach is not completely satisfactory. . . For example the fuzzy real line fails to be L-sober.

S.E. Rodabaugh, Point set lattice theoretic topology, Fuzzy Sets and Systems 40 (1991) 296–345.

On lattice-valued frames Liu and Zhang’s approach

On lattice-valued frames Then L can be embedded in a natural way into τ:

iX : L → τ, iX (α) = α for each α ∈ L.

iX is a frame embedding. If f :(X, τ) → (Y , σ) is an L-continuous map between two stratiﬁed L-topological spaces, then f −1 : σ → τ is a frame homomorphism such that −1 iY = f ◦ iX .

L-valued frames previous approaches (II)

Let L be a frame and (X, τ) a stratiﬁed L-topological space.

D. Zhan, Y. Liu, A localic L-fuzzy modiﬁcation of topologoical spaces, Fuzzy Sets and Systems 56 (1993) 215-227.

D. Zhan, Y. Liu, L-fuzzy version of Stone’s representation theorem for distributive lattices, Fuzzy Sets and Systems 76 (1995) 259-270.

On lattice-valued frames iX is a frame embedding. If f :(X, τ) → (Y , σ) is an L-continuous map between two stratiﬁed L-topological spaces, then f −1 : σ → τ is a frame homomorphism such that −1 iY = f ◦ iX .

L-valued frames previous approaches (II)

Let L be a frame and (X, τ) a stratiﬁed L-topological space. Then L can be embedded in a natural way into τ:

iX : L → τ, iX (α) = α for each α ∈ L.

D. Zhan, Y. Liu, A localic L-fuzzy modiﬁcation of topologoical spaces, Fuzzy Sets and Systems 56 (1993) 215-227.

D. Zhan, Y. Liu, L-fuzzy version of Stone’s representation theorem for distributive lattices, Fuzzy Sets and Systems 76 (1995) 259-270.

On lattice-valued frames If f :(X, τ) → (Y , σ) is an L-continuous map between two stratiﬁed L-topological spaces, then f −1 : σ → τ is a frame homomorphism such that −1 iY = f ◦ iX .

L-valued frames previous approaches (II)

Let L be a frame and (X, τ) a stratiﬁed L-topological space. Then L can be embedded in a natural way into τ:

iX : L → τ, iX (α) = α for each α ∈ L.

iX is a frame embedding.

D. Zhan, Y. Liu, A localic L-fuzzy modiﬁcation of topologoical spaces, Fuzzy Sets and Systems 56 (1993) 215-227.

D. Zhan, Y. Liu, L-fuzzy version of Stone’s representation theorem for distributive lattices, Fuzzy Sets and Systems 76 (1995) 259-270.

On lattice-valued frames L-valued frames previous approaches (II)

Let L be a frame and (X, τ) a stratiﬁed L-topological space. Then L can be embedded in a natural way into τ:

iX : L → τ, iX (α) = α for each α ∈ L.

iX is a frame embedding. If f :(X, τ) → (Y , σ) is an L-continuous map between two stratiﬁed L-topological spaces, then f −1 : σ → τ is a frame homomorphism such that −1 iY = f ◦ iX .

D. Zhan, Y. Liu, A localic L-fuzzy modiﬁcation of topologoical spaces, Fuzzy Sets and Systems 56 (1993) 215-227.

D. Zhan, Y. Liu, L-fuzzy version of Stone’s representation theorem for distributive lattices, Fuzzy Sets and Systems 76 (1995) 259-270.

On lattice-valued frames A continuous map between two L-fuzzy locales f :(A, iA) → (B, iB) is a frame homomorphism f : B → A such that

iB = f ◦ iA.

The category consisting of L-fuzzy locales as objects and continuous maps as morphism is called the category of L-fuzzy locales, which is easily checked to be the comma category Loc ↓ L.

L-valued frames previous approaches (II)

An L-fuzzy locale is deﬁned to be a pair (A, iA), where A is a frame and iA : L → A is a frame homomorphism.

D. Zhan, Y. Liu, L-fuzzy version of Stone’s representation theorem for distributive lattices, Fuzzy Sets and Systems 76 (1995) 259-270.

On lattice-valued frames The category consisting of L-fuzzy locales as objects and continuous maps as morphism is called the category of L-fuzzy locales, which is easily checked to be the comma category Loc ↓ L.

L-valued frames previous approaches (II)

An L-fuzzy locale is deﬁned to be a pair (A, iA), where A is a frame and iA : L → A is a frame homomorphism.

A continuous map between two L-fuzzy locales f :(A, iA) → (B, iB) is a frame homomorphism f : B → A such that

iB = f ◦ iA.

D. Zhan, Y. Liu, L-fuzzy version of Stone’s representation theorem for distributive lattices, Fuzzy Sets and Systems 76 (1995) 259-270.

On lattice-valued frames L-valued frames previous approaches (II)

An L-fuzzy locale is deﬁned to be a pair (A, iA), where A is a frame and iA : L → A is a frame homomorphism.

A continuous map between two L-fuzzy locales f :(A, iA) → (B, iB) is a frame homomorphism f : B → A such that

iB = f ◦ iA.

The category consisting of L-fuzzy locales as objects and continuous maps as morphism is called the category of L-fuzzy locales, which is easily checked to be the comma category Loc ↓ L.

D. Zhan, Y. Liu, L-fuzzy version of Stone’s representation theorem for distributive lattices, Fuzzy Sets and Systems 76 (1995) 259-270.

On lattice-valued frames • Deﬁnition

• Motivation: the iota functor ιL

Chain-valued frames

On lattice-valued frames • Deﬁnition

Chain-valued frames

• Motivation: the iota functor ιL

On lattice-valued frames Chain-valued frames

• Motivation: the iota functor ιL

• Deﬁnition

On lattice-valued frames Let L be a complete lattice and X be a set. For a ﬁxed α ∈ L \{1} and let a ∈ LX , we denote

[a 6≤ α] = {x ∈ X : a(x) 6≤ α}.

X X This deﬁnes a map ια : L → 2 by ια(a) = [a 6≤ α]. Now, given an L-topology τ on X, we consider the topology

ιL(τ) = h{ια(τ): α ∈ L}i = h{ια(a): a ∈ τ, α ∈ L}i.

This deﬁnes a functor ιL : L-Top → Top by

ιL(X, τ) = (X, ιL(τ)), ιL(h) = h.

The ιL functor

The iota functor ιL was originally introduced by Lowen with L = [0, 1] and later on extended by Kubiak to an arbitrary complete lattice.

On lattice-valued frames Now, given an L-topology τ on X, we consider the topology

ιL(τ) = h{ια(τ): α ∈ L}i = h{ια(a): a ∈ τ, α ∈ L}i.

This deﬁnes a functor ιL : L-Top → Top by

ιL(X, τ) = (X, ιL(τ)), ιL(h) = h.

The ιL functor

The iota functor ιL was originally introduced by Lowen with L = [0, 1] and later on extended by Kubiak to an arbitrary complete lattice. Let L be a complete lattice and X be a set. For a ﬁxed α ∈ L \{1} and let a ∈ LX , we denote

[a 6≤ α] = {x ∈ X : a(x) 6≤ α}.

X X This deﬁnes a map ια : L → 2 by ια(a) = [a 6≤ α].

On lattice-valued frames This deﬁnes a functor ιL : L-Top → Top by

ιL(X, τ) = (X, ιL(τ)), ιL(h) = h.

The ιL functor

[a 6≤ α] = {x ∈ X : a(x) 6≤ α}.

X X This deﬁnes a map ια : L → 2 by ια(a) = [a 6≤ α]. Now, given an L-topology τ on X, we consider the topology

ιL(τ) = h{ια(τ): α ∈ L}i = h{ια(a): a ∈ τ, α ∈ L}i.

On lattice-valued frames The ιL functor

[a 6≤ α] = {x ∈ X : a(x) 6≤ α}.

X X This deﬁnes a map ια : L → 2 by ια(a) = [a 6≤ α]. Now, given an L-topology τ on X, we consider the topology

ιL(τ) = h{ια(τ): α ∈ L}i = h{ια(a): a ∈ τ, α ∈ L}i.

This deﬁnes a functor ιL : L-Top → Top by

ιL(X, τ) = (X, ιL(τ)), ιL(h) = h.

On lattice-valued frames S (F1) ιL(τ) = h α∈L\{1} ια(τ)i. (collectionwise extremally epimorphic)

(F2) If a 6= b in τ, then ια(a) 6= ια(b) for some α ∈ L \{1}. (collectionwise monomorphic)

We can consider the system of frame homomorphisms ια : τ → ιL(τ) | α ∈ L \{1} .

The following are satisﬁed:

(F0) For each ∅ 6= S ⊆ L \{1}, we have that V S S ιV S(a) = [a > S] = [a > α] = ια(a). α∈S α∈S

The ιL functor chain valued frames

Let L be a complete chain. Then ια(a) = [a > α].

On lattice-valued frames S (F1) ιL(τ) = h α∈L\{1} ια(τ)i. (collectionwise extremally epimorphic)

(F2) If a 6= b in τ, then ια(a) 6= ια(b) for some α ∈ L \{1}. (collectionwise monomorphic)

(F0) For each ∅ 6= S ⊆ L \{1}, we have that V S S ιV S(a) = [a > S] = [a > α] = ια(a). α∈S α∈S

The ιL functor chain valued frames

Let L be a complete chain. Then ια(a) = [a > α].

We can consider the system of frame homomorphisms ια : τ → ιL(τ) | α ∈ L \{1} .

The following are satisﬁed:

On lattice-valued frames S (F1) ιL(τ) = h α∈L\{1} ια(τ)i. (collectionwise extremally epimorphic)

(F2) If a 6= b in τ, then ια(a) 6= ια(b) for some α ∈ L \{1}. (collectionwise monomorphic)

The ιL functor chain valued frames

Let L be a complete chain. Then ια(a) = [a > α].

We can consider the system of frame homomorphisms ια : τ → ιL(τ) | α ∈ L \{1} .

The following are satisﬁed:

(F0) For each ∅ 6= S ⊆ L \{1}, we have that V S S ιV S(a) = [a > S] = [a > α] = ια(a). α∈S α∈S

On lattice-valued frames S (F1) ιL(τ) = h α∈L\{1} ια(τ)i. (collectionwise extremally epimorphic)

(F2) If a 6= b in τ, then ια(a) 6= ια(b) for some α ∈ L \{1}. (collectionwise monomorphic)

The ιL functor chain valued frames

Let L be a complete chain. Then ια(a) = [a > α].

We can consider the system of frame homomorphisms ια : τ → ιL(τ) | α ∈ L \{1} .

The following are satisﬁed:

(F0) For each ∅ 6= S ⊆ L \{1}, we have that V S S ιV S(a) = [a > S] = [a > α] = ια(a). α∈S α∈S

On lattice-valued frames (F2) If a 6= b in τ, then ια(a) 6= ια(b) for some α ∈ L \{1}. (collectionwise monomorphic)

The ιL functor chain valued frames

Let L be a complete chain. Then ια(a) = [a > α].

We can consider the system of frame homomorphisms ια : τ → ιL(τ) | α ∈ L \{1} .

The following are satisﬁed:

(F0) For each ∅ 6= S ⊆ L \{1}, we have that V S S ιV S(a) = [a > S] = [a > α] = ια(a). α∈S α∈S S (F1) ιL(τ) = h α∈L\{1} ια(τ)i. (collectionwise extremally epimorphic)

On lattice-valued frames The ιL functor chain valued frames

Let L be a complete chain. Then ια(a) = [a > α].

We can consider the system of frame homomorphisms ια : τ → ιL(τ) | α ∈ L \{1} .

The following are satisﬁed:

(F0) For each ∅ 6= S ⊆ L \{1}, we have that V S S ιV S(a) = [a > S] = [a > α] = ια(a). α∈S α∈S S (F1) ιL(τ) = h α∈L\{1} ια(τ)i. (collectionwise extremally epimorphic)

(F2) If a 6= b in τ, then ια(a) 6= ια(b) for some α ∈ L \{1}. (collectionwise monomorphic)

On lattice-valued frames 2 T = ιL(τ),

3 ια ◦ κ = ϕα for each α ∈ L \{1}.

Then there is a frame τ and an isomorphism κ : A → τ such that:

1 (X, τ) is an L-space,

The above discussion means that A is, up to frame isomorphism, an L-topology on X.

The ιL functor chain valued frames

Let L be a complete chain, A a frame and (X, T ) a topological space and let ϕα : A → T | α ∈ L \{1} be a system of frame homomorphisms satisfying (F0), (F1) and (F2).

On lattice-valued frames 2 T = ιL(τ),

3 ια ◦ κ = ϕα for each α ∈ L \{1}.

1 (X, τ) is an L-space,

The above discussion means that A is, up to frame isomorphism, an L-topology on X.

The ιL functor chain valued frames

Let L be a complete chain, A a frame and (X, T ) a topological space and let ϕα : A → T | α ∈ L \{1} be a system of frame homomorphisms satisfying (F0), (F1) and (F2).

Then there is a frame τ and an isomorphism κ : A → τ such that:

On lattice-valued frames 2 T = ιL(τ),

3 ια ◦ κ = ϕα for each α ∈ L \{1}.

The above discussion means that A is, up to frame isomorphism, an L-topology on X.

The ιL functor chain valued frames

Let L be a complete chain, A a frame and (X, T ) a topological space and let ϕα : A → T | α ∈ L \{1} be a system of frame homomorphisms satisfying (F0), (F1) and (F2).

Then there is a frame τ and an isomorphism κ : A → τ such that:

1 (X, τ) is an L-space,

On lattice-valued frames 3 ια ◦ κ = ϕα for each α ∈ L \{1}.

The above discussion means that A is, up to frame isomorphism, an L-topology on X.

The ιL functor chain valued frames

Let L be a complete chain, A a frame and (X, T ) a topological space and let ϕα : A → T | α ∈ L \{1} be a system of frame homomorphisms satisfying (F0), (F1) and (F2).

Then there is a frame τ and an isomorphism κ : A → τ such that:

1 (X, τ) is an L-space,

2 T = ιL(τ),

On lattice-valued frames The above discussion means that A is, up to frame isomorphism, an L-topology on X.

The ιL functor chain valued frames

Let L be a complete chain, A a frame and (X, T ) a topological space and let ϕα : A → T | α ∈ L \{1} be a system of frame homomorphisms satisfying (F0), (F1) and (F2).

Then there is a frame τ and an isomorphism κ : A → τ such that:

1 (X, τ) is an L-space,

2 T = ιL(τ),

3 ια ◦ κ = ϕα for each α ∈ L \{1}.

On lattice-valued frames The ιL functor chain valued frames

Let L be a complete chain, A a frame and (X, T ) a topological space and let ϕα : A → T | α ∈ L \{1} be a system of frame homomorphisms satisfying (F0), (F1) and (F2).

Then there is a frame τ and an isomorphism κ : A → τ such that:

1 (X, τ) is an L-space,

2 T = ιL(τ),

3 ια ◦ κ = ϕα for each α ∈ L \{1}.

The above discussion means that A is, up to frame isomorphism, an L-topology on X.

On lattice-valued frames Chain valued frames

“The notion of chain-valued frame, is introduced to be an abstraction of the distinctive properties of the system of level mappings from an L-topology τ into ιL(τ). These conditions, when L is a complete chain, were taken as axioms (F0), (F1) and (F2) in order to deﬁne L-frames and the associated category L-Frm.”

A. Pultr, S.E. Rodabaugh, Lattice-valued frames, functor categories, and classes of sober spaces, in: Topological and Algebraic Structures in Fuzzy Sets: A Handbook of Recent Developments in the Mathematics of Fuzzy Sets, Kluwer Academic Publishers, 2003, pp. 153–187, (Chapter 6).

On lattice-valued frames l S A u (F1) A = h α∈L\{1} ϕα(A )i. (collectionwise extremally epimorphic)

u A A (F2) If a 6= b in A , then ϕα(a) 6= ϕα(b) for some α ∈ L \{1}. (collectionwise monomorphic)

A W A (F0) ϕV S = α∈S ϕα for every ∅ 6= S ⊆ L \{1}.

Chain valued frames

Let L be a complete chain. An L-frame A is a system

A u l ϕα : A → A α ∈ L \{1}

of frame morphisms – Au is the upper frame and Al is the lower frame – satisfying each of these conditions:

On lattice-valued frames l S A u (F1) A = h α∈L\{1} ϕα(A )i. (collectionwise extremally epimorphic)

u A A (F2) If a 6= b in A , then ϕα(a) 6= ϕα(b) for some α ∈ L \{1}. (collectionwise monomorphic)

Chain valued frames

Let L be a complete chain. An L-frame A is a system

A u l ϕα : A → A α ∈ L \{1}

of frame morphisms – Au is the upper frame and Al is the lower frame – satisfying each of these conditions:

A W A (F0) ϕV S = α∈S ϕα for every ∅ 6= S ⊆ L \{1}.

On lattice-valued frames u A A (F2) If a 6= b in A , then ϕα(a) 6= ϕα(b) for some α ∈ L \{1}. (collectionwise monomorphic)

Chain valued frames

Let L be a complete chain. An L-frame A is a system

A u l ϕα : A → A α ∈ L \{1}

of frame morphisms – Au is the upper frame and Al is the lower frame – satisfying each of these conditions:

A W A (F0) ϕV S = α∈S ϕα for every ∅ 6= S ⊆ L \{1}.

l S A u (F1) A = h α∈L\{1} ϕα(A )i. (collectionwise extremally epimorphic)

On lattice-valued frames Chain valued frames

Let L be a complete chain. An L-frame A is a system

A u l ϕα : A → A α ∈ L \{1}

of frame morphisms – Au is the upper frame and Al is the lower frame – satisfying each of these conditions:

A W A (F0) ϕV S = α∈S ϕα for every ∅ 6= S ⊆ L \{1}.

l S A u (F1) A = h α∈L\{1} ϕα(A )i. (collectionwise extremally epimorphic)

u A A (F2) If a 6= b in A , then ϕα(a) 6= ϕα(b) for some α ∈ L \{1}. (collectionwise monomorphic)

On lattice-valued frames such that the following diagram is commutative for each α ∈ L \{1}

A ϕα Au / Al

hu hl

B ϕα Bu / Bl

The resulting category, with composition and identities component-wise in Frm, is denoted by L-Frm.

Chain valued frames

An L-frame morphism h : A → B is an ordered pair of frame homomorphisms

hu : Au → Bu and hl : Al → Bl

On lattice-valued frames The resulting category, with composition and identities component-wise in Frm, is denoted by L-Frm.

Chain valued frames

An L-frame morphism h : A → B is an ordered pair of frame homomorphisms

hu : Au → Bu and hl : Al → Bl

such that the following diagram is commutative for each α ∈ L \{1}

A ϕα Au / Al

hu hl

B ϕα Bu / Bl

On lattice-valued frames Chain valued frames

An L-frame morphism h : A → B is an ordered pair of frame homomorphisms

hu : Au → Bu and hl : Al → Bl

such that the following diagram is commutative for each α ∈ L \{1}

A ϕα Au / Al

hu hl

B ϕα Bu / Bl

The resulting category, with composition and identities component-wise in Frm, is denoted by L-Frm.

On lattice-valued frames “During the preparation of the Volume, U. Höhle communicated to the authors of Chapter 6 that a complete chain is really only needed for its meet-irreducibles, and that for spatial L one also has meet-irreducibles which sufﬁce for the constructions of Chapter 6.”

U. Höhle and S.E. Rodabaugh, Weakening the requirement that L be a complete chain, in: Topological and Algebraic Structures in Fuzzy Sets: A Handbook of Recent Developments in the Mathematics of Fuzzy Sets, Kluwer Academic Publishers, 2003, pp. 189–197, (Chapter 7).

Chain valued frames

In the previous deﬁnitions L is assumed to be a complete chain, which seems to be quite a restrictive assumption.

On lattice-valued frames Chain valued frames

In the previous deﬁnitions L is assumed to be a complete chain, which seems to be quite a restrictive assumption.

“During the preparation of the Volume, U. Höhle communicated to the authors of Chapter 6 that a complete chain is really only needed for its meet-irreducibles, and that for spatial L one also has meet-irreducibles which sufﬁce for the constructions of Chapter 6.”

On lattice-valued frames • Lattice-valued frames for CD lattices

• Completely distributive lattices

The completely distributive case

On lattice-valued frames • Lattice-valued frames for CD lattices

The completely distributive case

• Completely distributive lattices

On lattice-valued frames The completely distributive case

• Completely distributive lattices

• Lattice-valued frames for CD lattices

On lattice-valued frames Completely distributive lattices

On lattice-valued frames Recall that L is continuous if and only if the way-below relation is approximating, i.e., if and only if

α = W{β ∈ L : β α} for each α ∈ L.

Let α, β, γ, δ ∈ L, then: (1) α β implies α ≤ β. (2) α ≤ β γ ≤ δ implies α δ. (3) If L is continuous, then α β implies α γ β for some γ ∈ L

Completely distributive lattices

Given α, β ∈ L, we say that α is way below β, in symbols α β, if and only if ) S ⊆ L and n =⇒ there exist γ , . . . γn ∈ S such that α ≤ ∨ γ . β ≤ W S 1 i=1 i

On lattice-valued frames Let α, β, γ, δ ∈ L, then: (1) α β implies α ≤ β. (2) α ≤ β γ ≤ δ implies α δ. (3) If L is continuous, then α β implies α γ β for some γ ∈ L

Completely distributive lattices

Given α, β ∈ L, we say that α is way below β, in symbols α β, if and only if ) S ⊆ L and n =⇒ there exist γ , . . . γn ∈ S such that α ≤ ∨ γ . β ≤ W S 1 i=1 i

Recall that L is continuous if and only if the way-below relation is approximating, i.e., if and only if

α = W{β ∈ L : β α} for each α ∈ L.

On lattice-valued frames Completely distributive lattices

Given α, β ∈ L, we say that α is way below β, in symbols α β, if and only if ) S ⊆ L and n =⇒ there exist γ , . . . γn ∈ S such that α ≤ ∨ γ . β ≤ W S 1 i=1 i

Recall that L is continuous if and only if the way-below relation is approximating, i.e., if and only if

α = W{β ∈ L : β α} for each α ∈ L.

Let α, β, γ, δ ∈ L, then: (1) α β implies α ≤ β. (2) α ≤ β γ ≤ δ implies α δ. (3) If L is continuous, then α β implies α γ β for some γ ∈ L

On lattice-valued frames Namely, given α, β ∈ L we have α <<<<β if ) S ⊆ L and n =⇒ there exist γ , . . . γn ∈ S such that ∧ γ ≤ β. V S ≤ α 1 i=1 i

For each α ∈ L we write ↑α = {β ∈ L : α <<<<β}. Then we have that Lop is alertcontinuous if and only if it satisﬁes V V α = {β ∈ L : α <<<<β} = ↑α for all α ∈ L. The following properties of the binary relation <<<< will be needed: (1) α <<<<β implies α ≤ β. (2) α ≤ β <<<<γ ≤ δ implies α <<<<δ.

Completely distributive lattices

We shall be particularly interested in the opposite relation of the way-below relation in the lattice Lop, denoted by <<<<.

On lattice-valued frames For each α ∈ L we write ↑α = {β ∈ L : α <<<<β}. Then we have that Lop is alertcontinuous if and only if it satisﬁes V V α = {β ∈ L : α <<<<β} = ↑α for all α ∈ L. The following properties of the binary relation <<<< will be needed: (1) α <<<<β implies α ≤ β. (2) α ≤ β <<<<γ ≤ δ implies α <<<<δ.

Completely distributive lattices

We shall be particularly interested in the opposite relation of the way-below relation in the lattice Lop, denoted by <<<<. Namely, given α, β ∈ L we have α <<<<β if ) S ⊆ L and n =⇒ there exist γ , . . . γn ∈ S such that ∧ γ ≤ β. V S ≤ α 1 i=1 i

On lattice-valued frames Then we have that Lop is alertcontinuous if and only if it satisﬁes V V α = {β ∈ L : α <<<<β} = ↑α for all α ∈ L. The following properties of the binary relation <<<< will be needed: (1) α <<<<β implies α ≤ β. (2) α ≤ β <<<<γ ≤ δ implies α <<<<δ.

Completely distributive lattices

For each α ∈ L we write ↑α = {β ∈ L : α <<<<β}.

On lattice-valued frames The following properties of the binary relation <<<< will be needed: (1) α <<<<β implies α ≤ β. (2) α ≤ β <<<<γ ≤ δ implies α <<<<δ.

Completely distributive lattices

For each α ∈ L we write ↑α = {β ∈ L : α <<<<β}. Then we have that Lop is alertcontinuous if and only if it satisﬁes V V α = {β ∈ L : α <<<<β} = ↑α for all α ∈ L.

On lattice-valued frames Completely distributive lattices

On lattice-valued frames We recall now the following result:

Let L be a complete lattice. Then the following are equivalent: (1) L is completely distributive. (2) L is a spatial frame and Lop is continuous.

Completely distributive lattices

A lattice is called completely distributive iff it is complete and for any familiy {xj,k : j ∈ J, k ∈ K (j)} in L the identity V W W V xj,k = xj,f (j) (CD) j∈J k∈K (j) f ∈M j∈J

holds, where M is the set of choice functions deﬁned on J with values f (j) ∈ K (j).

On lattice-valued frames Completely distributive lattices

A lattice is called completely distributive iff it is complete and for any familiy {xj,k : j ∈ J, k ∈ K (j)} in L the identity V W W V xj,k = xj,f (j) (CD) j∈J k∈K (j) f ∈M j∈J

holds, where M is the set of choice functions deﬁned on J with values f (j) ∈ K (j).

We recall now the following result:

Let L be a complete lattice. Then the following are equivalent: (1) L is completely distributive. (2) L is a spatial frame and Lop is continuous.

On lattice-valued frames i.e. α = V (↑α ∩ Spec L) for each α ∈ L,

Completely distributive lattices

Let L be a complete lattice. Then the following are equivalent:

(1) L is completely distributive.

(2) L satisﬁes the following two properties:

(i) L is a spatial frame,

(ii) Lop is continuous.

On lattice-valued frames Completely distributive lattices

Let L be a complete lattice. Then the following are equivalent:

(1) L is completely distributive.

(2) L satisﬁes the following two properties:

(i) L is a spatial frame, i.e. α = V (↑α ∩ Spec L) for each α ∈ L, (ii) Lop is continuous.

On lattice-valued frames Completely distributive lattices

Let L be a complete lattice. Then the following are equivalent:

(1) L is completely distributive. (2) L satisﬁes the following two properties:

(i) L is a spatial frame, i.e. α = V (↑α ∩ Spec L) for each α ∈ L, (ii) Lop is continuous, i.e. V α = ↑α for all α ∈ L.

On lattice-valued frames Completely distributive lattices

Let L be a complete lattice. Then the following are equivalent:

(1) L is completely distributive. (2) L satisﬁes the following two properties:

(i) L is a spatial frame, i.e. α = V (↑α ∩ Spec L) for each α ∈ L, V (ii) p = ↑p ∩ Spec L for each p ∈ Spec L.

On lattice-valued frames Lattice-valued frames for CD lattices

On lattice-valued frames The ιL functor completely distributive lattices

Let L be a completely distributive lattice. The mapping ιp : τ → ιL(τ) is a frame morphism for each p ∈ Spec L (this is not true in general if p fails to be prime). Consider the system of frame morphisms ιp : τ → ιL(τ) | p ∈ Spec L .

On lattice-valued frames The ιL functor completely distributive lattices

V (F0)’ Since p = ↑p ∩ Spec L for each p ∈ Spec L, we have that

[f 6≤ p] = S [f 6≤ q] for each p ∈ Spec L and f ∈ LX . q∈↑↑p∩Spec L

Consequently, for each p ∈ Spec L, W ιp = ιq. q∈↑↑p∩Spec L

On lattice-valued frames The ιL functor completely distributive lattices

(F1) Since L is a spatial frame then {ιp(f ): f ∈ τ, p ∈ Spec L} is a subbase of ιL(τ). Indeed, for each α ∈ L we have α = V (↑α ∩ Spec L) and so S S ια(f ) = [f 6≤ α] = [f 6≤ p] = ιp(f ). p∈↑α∩Spec L p∈↑α∩Spec L

Hence, D S E ιL(τ) = ιp(τ) . p∈Spec L

On lattice-valued frames The ιL functor completely distributive lattices

(F2) Since L is a spatial frame, for each distinct a, b ∈ τ there exists x ∈ X such that a(x) 6= b(x), hence there exists p ∈ Spec L such that either f (x) ≤ p and g(x) 6≤ p or a(x) 6≤ p and b(x) ≤ p and so [a 6≤ p] 6= [b 6≤ p]. It follows that

if a 6= b ∈ τ then ιp(a) 6= ιp(b) for some p ∈ Spec L.

On lattice-valued frames The ιL functor completely distributive lattices

W (F0)’ ιp = q∈↑↑p∩Spec L ιq for each p ∈ Spec L.

DS E (F1) ιL(τ) = p∈Spec L ιp(τ) . (collectionwise extremally epimorphic)

(F2) If a 6= b in τ then ιp(a) 6= ιp(b) for some p ∈ Spec L. (collectionwise monomorphic)

On lattice-valued frames 2 T = ιL(τ),

3 ιp ◦ κ = ϕp for each p ∈ Spec L.

Then there is a frame τ and an isomorphism κ : A → τ such that:

1 (X, τ) is an L-space,

The above discussion means that A is, up to frame isomorphism, an L-topology on X.

The ιL functor completely distributive lattices

Let L be a completely distributive lattice, A a frame and (X, T ) a topological space and let ϕp : A → OX | p ∈ Spec L

be a system of frame homomorphisms satisfying (F0)’, (F1) and (F2).

On lattice-valued frames 2 T = ιL(τ),

3 ιp ◦ κ = ϕp for each p ∈ Spec L.

1 (X, τ) is an L-space,

The above discussion means that A is, up to frame isomorphism, an L-topology on X.

The ιL functor completely distributive lattices

Let L be a completely distributive lattice, A a frame and (X, T ) a topological space and let ϕp : A → OX | p ∈ Spec L

be a system of frame homomorphisms satisfying (F0)’, (F1) and (F2). Then there is a frame τ and an isomorphism κ : A → τ such that:

On lattice-valued frames 2 T = ιL(τ),

3 ιp ◦ κ = ϕp for each p ∈ Spec L.

The above discussion means that A is, up to frame isomorphism, an L-topology on X.

The ιL functor completely distributive lattices

Let L be a completely distributive lattice, A a frame and (X, T ) a topological space and let ϕp : A → OX | p ∈ Spec L

be a system of frame homomorphisms satisfying (F0)’, (F1) and (F2). Then there is a frame τ and an isomorphism κ : A → τ such that:

1 (X, τ) is an L-space,

On lattice-valued frames 3 ιp ◦ κ = ϕp for each p ∈ Spec L.

The above discussion means that A is, up to frame isomorphism, an L-topology on X.

The ιL functor completely distributive lattices

Let L be a completely distributive lattice, A a frame and (X, T ) a topological space and let ϕp : A → OX | p ∈ Spec L

be a system of frame homomorphisms satisfying (F0)’, (F1) and (F2). Then there is a frame τ and an isomorphism κ : A → τ such that:

1 (X, τ) is an L-space,

2 T = ιL(τ),

On lattice-valued frames The above discussion means that A is, up to frame isomorphism, an L-topology on X.

The ιL functor completely distributive lattices

Let L be a completely distributive lattice, A a frame and (X, T ) a topological space and let ϕp : A → OX | p ∈ Spec L

be a system of frame homomorphisms satisfying (F0)’, (F1) and (F2). Then there is a frame τ and an isomorphism κ : A → τ such that:

1 (X, τ) is an L-space,

2 T = ιL(τ),

3 ιp ◦ κ = ϕp for each p ∈ Spec L.

On lattice-valued frames The ιL functor completely distributive lattices

Let L be a completely distributive lattice, A a frame and (X, T ) a topological space and let ϕp : A → OX | p ∈ Spec L

be a system of frame homomorphisms satisfying (F0)’, (F1) and (F2). Then there is a frame τ and an isomorphism κ : A → τ such that:

1 (X, τ) is an L-space,

2 T = ιL(τ),

3 ιp ◦ κ = ϕp for each p ∈ Spec L.

The above discussion means that A is, up to frame isomorphism, an L-topology on X.

On lattice-valued frames In a complete chain L any non-empty subset is downdirected and A u l Spec L = L \{1} and hence ϕp : A → A | p ∈ Spec L is a system of frame morphisms, then axiom (F0) can be equivalently stated as:

A W A (F0) ϕV S = ϕs for each downdirected ∅ 6= S ⊆ Spec L. s∈S

For a completely distributive lattice L the following are equivalent

A W A (F0) ϕV S = s∈S ϕs for each downdirected ∅ 6= S ⊆ Spec L.

A W A (F0)’ ϕp = q∈↑↑p∩Spec L ϕq for each p ∈ Spec L.

The ιL functor completely distributive lattices

A subset S of a complete lattice is downdirected if it is non-empty and for any α, β ∈ S there is some γ ∈ S such that γ ≤ α ∧ β.

On lattice-valued frames For a completely distributive lattice L the following are equivalent

A W A (F0) ϕV S = s∈S ϕs for each downdirected ∅ 6= S ⊆ Spec L.

A W A (F0)’ ϕp = q∈↑↑p∩Spec L ϕq for each p ∈ Spec L.

The ιL functor completely distributive lattices

A subset S of a complete lattice is downdirected if it is non-empty and for any α, β ∈ S there is some γ ∈ S such that γ ≤ α ∧ β.

In a complete chain L any non-empty subset is downdirected and A u l Spec L = L \{1} and hence ϕp : A → A | p ∈ Spec L is a system of frame morphisms, then axiom (F0) can be equivalently stated as:

A W A (F0) ϕV S = ϕs for each downdirected ∅ 6= S ⊆ Spec L. s∈S

On lattice-valued frames The ιL functor completely distributive lattices

A subset S of a complete lattice is downdirected if it is non-empty and for any α, β ∈ S there is some γ ∈ S such that γ ≤ α ∧ β.

A W A (F0) ϕV S = ϕs for each downdirected ∅ 6= S ⊆ Spec L. s∈S

For a completely distributive lattice L the following are equivalent

A W A (F0) ϕV S = s∈S ϕs for each downdirected ∅ 6= S ⊆ Spec L.

A W A (F0)’ ϕp = q∈↑↑p∩Spec L ϕq for each p ∈ Spec L.

On lattice-valued frames l S A u (F1) A = h p∈Spec L ϕp (A )i. (collectionwise extremally epimorphic)

u A A (F2) If a 6= b in A then ϕp (a) 6= ϕp (b) for some p ∈ Spec L. (collectionwise monomorphic)

A W A (F0) ϕp = q∈↑↑p∩Spec L ϕq for every p ∈ Spec L.

L-valued frames Let L be a completely distributive lattice. An L-frame A is a system

A u l ϕp : A → A p ∈ Spec L

of frame morphisms – Au is the upper frame and Al is the lower frame – satisfying each of these conditions:

J.G.G., U. Höhle, M.A. de Prada Vicente, On lattice-valued frames: the completely distributive case, Fuzzy Sets and Systems 159 (2010) 1022–1030.

On lattice-valued frames l S A u (F1) A = h p∈Spec L ϕp (A )i. (collectionwise extremally epimorphic)

u A A (F2) If a 6= b in A then ϕp (a) 6= ϕp (b) for some p ∈ Spec L. (collectionwise monomorphic)

L-valued frames Let L be a completely distributive lattice. An L-frame A is a system

A u l ϕp : A → A p ∈ Spec L

of frame morphisms – Au is the upper frame and Al is the lower frame – satisfying each of these conditions:

A W A (F0) ϕp = q∈↑↑p∩Spec L ϕq for every p ∈ Spec L.

J.G.G., U. Höhle, M.A. de Prada Vicente, On lattice-valued frames: the completely distributive case, Fuzzy Sets and Systems 159 (2010) 1022–1030.

On lattice-valued frames u A A (F2) If a 6= b in A then ϕp (a) 6= ϕp (b) for some p ∈ Spec L. (collectionwise monomorphic)

L-valued frames Let L be a completely distributive lattice. An L-frame A is a system

A u l ϕp : A → A p ∈ Spec L

of frame morphisms – Au is the upper frame and Al is the lower frame – satisfying each of these conditions:

A W A (F0) ϕp = q∈↑↑p∩Spec L ϕq for every p ∈ Spec L.

l S A u (F1) A = h p∈Spec L ϕp (A )i. (collectionwise extremally epimorphic)

J.G.G., U. Höhle, M.A. de Prada Vicente, On lattice-valued frames: the completely distributive case, Fuzzy Sets and Systems 159 (2010) 1022–1030.

On lattice-valued frames L-valued frames Let L be a completely distributive lattice. An L-frame A is a system

A u l ϕp : A → A p ∈ Spec L

of frame morphisms – Au is the upper frame and Al is the lower frame – satisfying each of these conditions:

A W A (F0) ϕp = q∈↑↑p∩Spec L ϕq for every p ∈ Spec L.

l S A u (F1) A = h p∈Spec L ϕp (A )i. (collectionwise extremally epimorphic)

u A A (F2) If a 6= b in A then ϕp (a) 6= ϕp (b) for some p ∈ Spec L. (collectionwise monomorphic)

J.G.G., U. Höhle, M.A. de Prada Vicente, On lattice-valued frames: the completely distributive case, Fuzzy Sets and Systems 159 (2010) 1022–1030.

On lattice-valued frames such that the following diagram is commutative for each p ∈ Spec L

A ϕp Au / Al

hu hl

ϕB p Bu / Bl

The resulting category, with composition and identities component-wise in Frm, is denoted by L-Frm.

L-valued frames

An L-frame morphism h : A → B is an ordered pair of frame homomorphisms

hu : Au → Bu and hl : Al → Bl

On lattice-valued frames The resulting category, with composition and identities component-wise in Frm, is denoted by L-Frm.

L-valued frames

An L-frame morphism h : A → B is an ordered pair of frame homomorphisms

hu : Au → Bu and hl : Al → Bl

such that the following diagram is commutative for each p ∈ Spec L

A ϕp Au / Al

hu hl

ϕB p Bu / Bl

On lattice-valued frames L-valued frames

An L-frame morphism h : A → B is an ordered pair of frame homomorphisms

hu : Au → Bu and hl : Al → Bl

such that the following diagram is commutative for each p ∈ Spec L

A ϕp Au / Al

hu hl

ϕB p Bu / Bl

The resulting category, with composition and identities component-wise in Frm, is denoted by L-Frm.

On lattice-valued frames • All the results proved by Pultr and Rodabaugh regarding the category L-Frm for a complete chain can now be extended to this new setting.

In particular if we denote by F0 and F1 the categories in which the objects are frame morphisms for which only (F0) (resp. (F0) and (F1)) is (resp. are) satisﬁed.Then

• F0 is complete and cocomplete and each of the forgetful functors u l U , U : F0 → Frm preserves all limits and colimits.

• F1 is complete and cocomplete. • L-Frm is complete and cocomplete.

L-valued frames

• The new notion coincides with that of Pultr and Rodabaugh when L is a complete chain.

On lattice-valued frames In particular if we denote by F0 and F1 the categories in which the objects are frame morphisms for which only (F0) (resp. (F0) and (F1)) is (resp. are) satisﬁed.Then

• F0 is complete and cocomplete and each of the forgetful functors u l U , U : F0 → Frm preserves all limits and colimits.

• F1 is complete and cocomplete. • L-Frm is complete and cocomplete.

L-valued frames

• The new notion coincides with that of Pultr and Rodabaugh when L is a complete chain.

• All the results proved by Pultr and Rodabaugh regarding the category L-Frm for a complete chain can now be extended to this new setting.

On lattice-valued frames • F0 is complete and cocomplete and each of the forgetful functors u l U , U : F0 → Frm preserves all limits and colimits.

• F1 is complete and cocomplete. • L-Frm is complete and cocomplete.

L-valued frames

• The new notion coincides with that of Pultr and Rodabaugh when L is a complete chain.

• All the results proved by Pultr and Rodabaugh regarding the category L-Frm for a complete chain can now be extended to this new setting.

In particular if we denote by F0 and F1 the categories in which the objects are frame morphisms for which only (F0) (resp. (F0) and (F1)) is (resp. are) satisﬁed.Then

On lattice-valued frames • F1 is complete and cocomplete. • L-Frm is complete and cocomplete.

L-valued frames

• The new notion coincides with that of Pultr and Rodabaugh when L is a complete chain.

• All the results proved by Pultr and Rodabaugh regarding the category L-Frm for a complete chain can now be extended to this new setting.

In particular if we denote by F0 and F1 the categories in which the objects are frame morphisms for which only (F0) (resp. (F0) and (F1)) is (resp. are) satisﬁed.Then

• F0 is complete and cocomplete and each of the forgetful functors u l U , U : F0 → Frm preserves all limits and colimits.

On lattice-valued frames • L-Frm is complete and cocomplete.

L-valued frames

• The new notion coincides with that of Pultr and Rodabaugh when L is a complete chain.

• All the results proved by Pultr and Rodabaugh regarding the category L-Frm for a complete chain can now be extended to this new setting.

In particular if we denote by F0 and F1 the categories in which the objects are frame morphisms for which only (F0) (resp. (F0) and (F1)) is (resp. are) satisﬁed.Then

• F0 is complete and cocomplete and each of the forgetful functors u l U , U : F0 → Frm preserves all limits and colimits.

• F1 is complete and cocomplete.

On lattice-valued frames L-valued frames

• The new notion coincides with that of Pultr and Rodabaugh when L is a complete chain.

• All the results proved by Pultr and Rodabaugh regarding the category L-Frm for a complete chain can now be extended to this new setting.

In particular if we denote by F0 and F1 the categories in which the objects are frame morphisms for which only (F0) (resp. (F0) and (F1)) is (resp. are) satisﬁed.Then

• F0 is complete and cocomplete and each of the forgetful functors u l U , U : F0 → Frm preserves all limits and colimits.

• F1 is complete and cocomplete. • L-Frm is complete and cocomplete.

On lattice-valued frames We have already speciﬁed an answer by proving that the condition of a complete chain can be relaxed to a completely distributive lattice and that the completeness and cocompleteness of F0, F1 and L-Frm are still satisﬁed. In this context the natural question arises whether weakening of complete distributivity is still possible. As an answer to this question we show that complete distributivity is necessary for the property that for every L-topological space (X, τ) the system ιp : τ → ιL(τ) | p ∈ Spec L

of frame homomorphisms ιp satisﬁes (F0) and (F2).

L-valued frames possible extensions

Our work was motivated by the question stated in the papers of Pultr and Rodabaugh, when the authors suggest that relaxing the condition of a complete chain is a signiﬁcant question.

On lattice-valued frames In this context the natural question arises whether weakening of complete distributivity is still possible. As an answer to this question we show that complete distributivity is necessary for the property that for every L-topological space (X, τ) the system ιp : τ → ιL(τ) | p ∈ Spec L

of frame homomorphisms ιp satisﬁes (F0) and (F2).

L-valued frames possible extensions

On lattice-valued frames As an answer to this question we show that complete distributivity is necessary for the property that for every L-topological space (X, τ) the system ιp : τ → ιL(τ) | p ∈ Spec L

of frame homomorphisms ιp satisﬁes (F0) and (F2).

L-valued frames possible extensions

Our work was motivated by the question stated in the papers of Pultr and Rodabaugh, when the authors suggest that relaxing the condition of a complete chain is a significant question. We have already specified an answer by proving that the condition of a complete chain can be relaxed to a completely distributive lattice and that the completeness and cocompleteness of F0, F1 and L-Frm are still satisfied. In this context the natural question arises whether weakening of complete distributivity is still possible.

On lattice-valued frames L-valued frames possible extensions

of frame homomorphisms ιp satisﬁes (F0) and (F2).

On lattice-valued frames • If (ιp)p∈Spec L satisﬁes axiom (F2) for each (X, τ), then L is spatial.

• If (ιp)p∈Spec L is an L-frame for each (X, τ), then L is a completely distributive lattice.

• If (ιp)p∈Spec L satisﬁes axiom (F0) for each (X, τ), then V p = ↑p ∩ Spec L for each p ∈ Spec L.

L-valued frames possible extensions

Let L be a frame, (X, τ) an L-topological space and ιp : τ → ιL(τ) | p ∈ Spec L

the system of frame morphisms determined by the ι-functor. Then:

On lattice-valued frames • If (ιp)p∈Spec L satisﬁes axiom (F2) for each (X, τ), then L is spatial.

• If (ιp)p∈Spec L is an L-frame for each (X, τ), then L is a completely distributive lattice.

L-valued frames possible extensions

Let L be a frame, (X, τ) an L-topological space and ιp : τ → ιL(τ) | p ∈ Spec L

the system of frame morphisms determined by the ι-functor. Then:

• If (ιp)p∈Spec L satisﬁes axiom (F0) for each (X, τ), then V p = ↑p ∩ Spec L for each p ∈ Spec L.

On lattice-valued frames • If (ιp)p∈Spec L is an L-frame for each (X, τ), then L is a completely distributive lattice.

L-valued frames possible extensions

Let L be a frame, (X, τ) an L-topological space and ιp : τ → ιL(τ) | p ∈ Spec L

the system of frame morphisms determined by the ι-functor. Then:

• If (ιp)p∈Spec L satisﬁes axiom (F0) for each (X, τ), then V p = ↑p ∩ Spec L for each p ∈ Spec L.

• If (ιp)p∈Spec L satisﬁes axiom (F2) for each (X, τ), then L is spatial.

On lattice-valued frames L-valued frames possible extensions

Let L be a frame, (X, τ) an L-topological space and ιp : τ → ιL(τ) | p ∈ Spec L

the system of frame morphisms determined by the ι-functor. Then:

• If (ιp)p∈Spec L satisﬁes axiom (F0) for each (X, τ), then V p = ↑p ∩ Spec L for each p ∈ Spec L.

• If (ιp)p∈Spec L satisﬁes axiom (F2) for each (X, τ), then L is spatial.

• If (ιp)p∈Spec L is an L-frame for each (X, τ), then L is a completely distributive lattice.

On lattice-valued frames Eskerrik asko!

¡Muchas gracias!

On lattice-valued frames