Continuous Monads: Continuous Lattices Revisited Ernie Manes, - - PDF document

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Continuous Monads: Continuous Lattices Revisited Ernie Manes, University of Massachusetts Amherst, Massachusetts, USA FMCS 2019 May 28, 2019

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1. Quick Overview 2. Lattices for Computation 3. Motivating Monads 4. Conditionals 5. Power Set Monads 6. Continuous Monads 7. The Two-Element Algebra 8. Scott Monads

1 QUICK OVERVIEW 3

1 Quick Overview Usually, datatypes are posets in which every directed set has a supre- mum. We generalize directed sets to ρ-sets and define ρ-continuous posets. Main theorem part 1: For every ρ there is a “continuous” monad Tρ whose algebras are the ρ-continuous posets. Main theorem part 2: Certain abstract axioms characterize whether a monad is of form Tρ for some ρ.

1 QUICK OVERVIEW 4

Every ρ-continous poset has two topologies, the Sierpi´ nski topology and the canonical topology. When ρ = directed we get continuous lattices with the Scott and Law- son topologies. For a Scott monad where each ρ-set is directed, the Sierpi´ nski topology is the Scott topology. The condition that a directed set be a ρ-set is necessary and sufficient for the canonical topology to be compact Hausdorff. For general ρ, an intersection of two ρ-Scott-open sets need not be ρ-Scott-open.

1 QUICK OVERVIEW 5

The vitality of the 1980 text Compendium of Continuous Lattices stems from the fact that

• Continuous lattices are data types (Scott topology).
• Continuous lattices are part of the theory of compact abelian semi-

groups (Lawson topology). Our aim is to keep this perspective in a wider range of structures. continuous ρ-continuous = multivariable calculus differential categories

2 LATTICES FOR COMPUTATION 6

2 Lattices for Computation In the early 70s, Dana Scott introduced the idea that computation states are partially ordered. x ≤ y means y has better information than x. The mathematical problem was to represent data types as lattices.

2 LATTICES FOR COMPUTATION 7

D ⊂ (X, ≤) is consistent if D = ∅ and if for a1, . . . , am ∈ D there exists a ∈ X with all ai ≤ a. Even stronger, D ⊂ (X, ≤) is directed if D = ∅ and if for a, b ∈ D there exists c ∈ D with a, b ≤ c. As a computation proceeds, we think of its intermediate states as begin consistent, hopefully directed. The final computation is the supremum.

2 LATTICES FOR COMPUTATION 8

Theorem In analysis, all topological spaces are Hausdorff. Proof Ask my analyst friends at UMass. Early topology texts were Alexandroff and Hopf [3] in 1935 and ˘ Cech [5] written in 1955 and later rewritten in English by Frolik and Kat˘ etov. Alexandroff and Hopf credited Kolmogoroff with the invention of T0

• spaces. These were studied in some depth by ˘

Cech.

2 LATTICES FOR COMPUTATION 9

Scott [19] pointed out that ˘ Cech [5, page 483] called T0 spaces “feebly semi-separated”. The Grothendieck school [11] noticed that if X is a T0 space, its “spe- cialization order” x ≤ y ⇔ x ∈ {y} yields a poset. There are many topologies whose specialization is a given poset. For such a topology

• {y}

= ↓y (= {x : x ≤ y})

• if x ≤ y, x ∈ (↓y)′, y /

∈ (↓y)′ so the topology is T0

• Each closed set A is a lower set, A =↓A.
• U is upper ⇔ U ′ is lower, so all open sets are upper sets.
• The topology is not T1 unless x ≤ y ⇒ x = y.

2 LATTICES FOR COMPUTATION 10

In his seminal paper [19], Dana Scott wanted a specialization topology

• n (X, ≤) such that when a directed set D is considered as a net in X,

it would have

D as one of its limits.

With this in mind, U ⊂ X is Scott open if

• U =↑U
• For directed D,

D ∈ U ⇒ U ∩ D = ∅

We say x is way below y, written x ≪ y, if whenever D is directed and y ≤

D then there exists d ∈ D with x ≤ d.

To explain this x ≤ y ⇔ x ∈ ↓y x ≪ y ⇔ x ∈ (↓y)o where (·)o is Scott interior.

2 LATTICES FOR COMPUTATION 11

First definition: a poset with directed suprema is continuous if the axiom of approximation holds, that is, x =

{y : y ≪ x}

There are many texts, e.g. Abramsky and Jung [1], and Gunter [12]. Note: The Lawson topology plays no role in these texts. Complete lattices are mathematically convenient, but it is problematic to interpret the greatest element, or arbitrary binary suprema. A set X of alternatives can be represented as the flat poset X ∪ {⊥} with ⊥≤ x. This is not a complete lattice. Additionally, the fundamental example of partial functions between an input set and an output set has non-empty infima (and hence bounded suprema), but does not have binary suprema.

2 LATTICES FOR COMPUTATION 12

Scott showed that the category of continuous lattices and morphisms which preserve directed suprema is cartesian-closed. This category isomorphically embeds (via the Scott topology) as the full subcategory of injective objects in T0 spaces and continuous maps. Morphisms cannot be strict because constants are continuous. The same paper had the D∞-construction: an arbitrary poset can be fully embedded in a continuous lattice which is isomorphic to its own function space. This provides the first model of the type-free lambda calculus.

2 LATTICES FOR COMPUTATION 13

While the Scott topology does not have enough open sets to be Haus- dorff, we can extend this topology to the Lawson topology by now allowing ↑x to be closed for each x. This topology is compact Hausdorff. A function between continous lattices is Lawson continuous if and only if it preserves all infima and directed suprema. This is a different and equally useful notion of morphism. So the Scott topology completely recovers the continuous lattice struc- ture ( x ≤ y ⇔ x ∈ {y} ). The Lawson topology identifies continuous lattices as a class of compact abelian semigroups. Hence the Compendium [8] has six authors.

3 MOTIVATING MONADS 14

3 Motivating Monads “Cayley theorem” for posets. (X, ≤) → (2X, ⊃), x → ↑x = {y : y ≥ x} x ≤ y ⇔ ↑x ⊃ ↑y We have already expressed interest in subsets of (X, ≤), such as con- sistent sets or directed sets. These will become families of subsets of X under the Cayley map. Starting point: BX = {A : A ⊂ 2X} is a monad in Set, the double- dualization monad induced by 2.

3 MOTIVATING MONADS 15

Say that two subsets A, D ⊂ (X, ≤) are mutually cofinal if ∀ a ∈ A ∃ d ∈ D a ≤ d ∀ d ∈ D ∃ a ∈ A d ≤ a For example, if (an) is a chain then (a2n) and (a3n) are mutually cofinal and have ther same supremum. It is natural to seek a version of the Cayley representation which identifies these.

3 MOTIVATING MONADS 16

With this in mind, consider these definitions:

• For A ∈ BX, Ac = {D ⊂ X : ∃ A ∈ A D ⊃ A}.

Note: For A ⊂ (2X, ⊃), Ac =↓A.

• BcX = {A ∈ BX : A = Ac}

Bc is a submonad of B. The familification map Υ for subsets of a poset is A ⊂ (X, ≤) → ΥA = {↑a : a ∈ A}c ∈ BcX So A, D are mutually cofinal ⇔ ΥA = ΥD. We always have ΥA = Υ(↓A).

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We are heading to a generalization ρ of “consistent” and “directed”. We call such a ρ a conditional (for suprema), and it assigns to each poset a collection of subsets called ρ-sets. Define TρX = {A ∈ BcX : A is a ρ-set in (2X, ⊃)} As we shall see, the axioms on ρ will force Tρ to be a submonad of Bc whose algebras are the ρ-continuous posets.

3 MOTIVATING MONADS 18

In my 1982 paper A class of fuzzy theories about Kleisli categories I noticed that NX = {F : F is a filter with non-empty intersection} was a submonad of the filter monad. I wondered what the algebras

• were. Trying to find out recently , I got started on all of this.

It turns out that N is a Scott monad with NX = TρX if a ρ-set is a directed and bounded set. The N-algebras are a reasonable category

• f data types, called continuous posets by some (e.g. [7, 17]). They

include

• Continuous lattices
• Flat posets
• Pfn(A, B)

I believe it is new that NX is the free continuous poset.

3 MOTIVATING MONADS 19

The monad point of view leads to the following fundamental diagram which has at least three useful interpretations: X TX 22X

✲ ❄

ξ

• 1. The two-element poset generates the whole variety.
• 2. The Sierpi´

nski topology on 2 induces the “Scott topology”.

• 3. The discrete topology on 2 induces the “Lawson topology”.

More details later.

3 MOTIVATING MONADS 20

Define L to be the category of posets with non-empty infima and mor- phisms which preserve these. L-splitting lemma: Surjective morphisms in L split.

• Proof. For surjective f define g

Y

g

− → X

f

− → Y by gy =

{x : fx = y}.

Has anybody seen this in the literature?

4 CONDITIONALS 21

4 Conditionals Before giving axioms for ρ-sets, let us set down the intended semantics. There are two versions, as follows. A ρ-poset has all non-empty infima and has suprema for all ρ-sets. Morphisms preserve non-empty infima and ρ-suprema. A ρ-poset is a ρ-poset with a greatest element. Morphisms must also preserve the greatest element.

4 CONDITIONALS 22

The ρ-below relation: In a ρ-poset, say x ≪ρ y ⇔ for D a ρ-set with

D ≤ y, ∃ d ∈ D with x ≤ d

Define a ρ-continuous poset to be a ρ-poset for which for all x x =

D, for some ρ-set D ⊂ {y : y ≪ρ x}

The semantics: Cρ is the full subcategory of ρ-continuous posets.

• Cρ is the full subcategory of ρ-continuous

ρ-posets. When ρ-set = directed set, the objects of Cρ are dcpo’s, and the objects of Cρ are continuous lattices.

4 CONDITIONALS 23

Here is a second (equivalent) definition of a ρ-continuous poset. The ρ-distributive law is

• j∈J
• k∈K(j)

xjk =

• f∈ K(j)
• j∈J

xj,fj (CDρ) Here it is understood that both suprema are ρ-suprema. Theorem A ρ-poset is ρ-continuous if and only if it is ρ-distributive. The proof is virtually the same as that for the special case of continuous lattices in the Compendium, or in Abramsky and Jung.

4 CONDITIONALS 24

Example Let every set be a ρ-set. Thus Tρ = Bc. The ρ-continuous posets are the completely-distributive lattices

4 CONDITIONALS 25

The axioms on ρ. (ρ.1, ρ.2, ρ.3, ρ.4, ρ.5, ρ.6) will lead to Cρ. (ρ.1, ρ.2, ρ.3, ρ.4, ρ.5, ρ.6) will lead to Cρ. (ρ.1) Every subset with a greatest element is a ρ-set. (ρ.2) The image of a ρ-set under an order-preserving map is a ρ-set. (ρ.3) If A is a ρ-set in (X, ≤), so is ↓A. (ρ.4) If Ai is a ρ-set in (Xi, ≤i) then

Ai is a ρ-set in (Xi, ≤i).

4 CONDITIONALS 26

For the remaining two axioms, define TρX = {A ∈ BcX : A is a ρ-set in (2X\{∅}, ⊃)}

• TρX = {A ∈ BcX : A is a ρ-set in (2X, ⊃)}

(ρ.5 | ρ.5) If {Ai : i ∈ I} is a ρ-set in TρX | TρX (under inclusion) then

• i Ai ∈ TρX |

TρX. If ρ is “directed”, this says every directed union of filters is a filter. (ρ.6 | ρ.6) If A ∈ TρX | TρX and Bx ∈ TρY | TρY for all x ∈ X then {D ⊂ Y : {x : D ∈ Bx} ∈ A} ∈ TρY | TρY

4 CONDITIONALS 27

Main Theorem Part 1 For any conditional ρ, Tρ is a monad whose category of algebras is Cρ. Similarly for Tρ and Cρ. We’ll say more about the proof later. Thus the Bc-algebras are completely-distributive lattices. (Markowsky had noted [17] that BcX is the free completely-distributive lattice gen- erated by X). When ρ-set = directed set, Tρ is the filter monad and we recover the theorem of Alan Day and Oswald Wyler (independently) that these are continuous lattices.

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5 Power Set Monads The power set monad P in Set is defined by PX = 2X For X

f

− → Y, (Pf)A = {fa : a ∈ A} ηX x = {x} PPX

µX

− − → PX, µX(A) =

A

For X

f

− → PY, f #A =

• a∈A

fa Recall: (·)# and (µ, T) are coextensive for any monad T. µX = (idTX)#, Tf = (ηY f)# For X

f

− → TY, f # = TX

Tf

− → TTY

µY

− → TY More generally, if f : X → (Y, θ), f # = TX

Tf

− → TY

θ

− → Y .

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If T = (T, η, (·)#) is a monad in a category and iX : SX → TX is monic as X ranges over all objects, then S is a submonad of T if

• For all X, ηX factors through SX.
• For X

f

− → SY , (X

f

− → SY

iX

− → TY )

#

maps SX into SY . Of course, a submonad S is a monad S in its own right.

5 POWER SET MONADS 30

Recall that an infinite cardinal κ is regular if it is not the union of fewer than κ sets each of cardinality < κ. Say that a submonad T of P is nontrivial if there exists X and A ∈ TX such that A has more than one element. Theorem Every nontrivial submonad of P is one of the following:

• 1. P
• 2. PoX = {A ⊂ X : A = ∅}
• 3. PκX = {A ⊂ X : |A| < κ}, κ regular
• 4. Pκ ∩ Po

Proof idea. For f : X → Y , (Pf)(A) = {fa : a ∈ A}. In Evelyn Nelson’s paper [18] on z-continuous algebras, z is a submonad

• f P.

5 POWER SET MONADS 31

What is unusual about the next theorem is that we can characterize the algebras for all submonads at once. Theorem Let T be a nontrivial submonad of P. Then a T-algebra is a partially-ordered set (X, ≤) for which A ⊂ X has an infimum if and

• nly if A ∈ TX. The structure map ξ : TX → X is

ξA =

A

The morphisms preserve such infima.

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6 Continuous Monads BX = 22X is a monad B = (B, prin, (·)#, µ) where, recall, prin(x) = {A ⊂ X : x ∈ A} For f : X → Y , (Bf)A = {D ⊂ Y : f −1D ∈ A For f : X → BY , f #(A) = {D ⊂ Y : {x : D ∈ fx} ∈ A} If for A ⊂ X ✷A = {A ∈ BX : A ∈ A} then µX(H) = {A ⊂ X : ✷A ∈ H}

6 CONTINUOUS MONADS 33

The B-algebras are complete, atomic Boolean algebras with morphisms that preserve infima, suprema and complements. These algebras satisfy an “axiom of approximation” in that every ele- ment is the supremum of the atoms beneath it. The structure ξ : BX → X is A →

{x : x is an atom, ↑x ∈ A}

(See [16]). A family monad is a submonad of B. Examples include Bc and F, FX = filters on X.

6 CONTINUOUS MONADS 34

Proposition P is a submonad of Bc Proof PX → BcX, A → Prin(A) = {B ⊂ X : B ⊃ A} is a monad map. Note: A ⊂ B ⇔ Prin(A) ⊃ Prin(B).

6 CONTINUOUS MONADS 35

Definition A continuous monad is a submonad T of Bc with Po ⊂ T ⊂ Bc such that (♣) if A ∈ TX then {2A : A ∈ A}c ∈ T(2X) (♣♣) if Ai ∈ TXi (i ∈ I) then {

Ai : Ai ∈ Ai}c ∈ T( Xi)

Open question: We don’t currently know an example of Po ⊂ T ⊂ Bc where either axiom fails.

6 CONTINUOUS MONADS 36

If T ⊂ Bc (e.g. T is a continuous monad) and A ∈ TX then A =

• A∈A

Prin(A) =

• A∈A
• x∈A

prin(x) In universal algebra, an element of the free algebra generated by X is an “expression” in the variables in X. Here, a “variable” is prin(x).

6 CONTINUOUS MONADS 37

Theorem For a conditional ρ, Tρ and Tρ are continuous monads. Proof comments. Recall TρX = {A ∈ BcX : A is a ρ-set in (2X\{∅}, ⊃)} If ∅ = A ⊂ X then, in (2X\{∅}, ⊃), Prin(A) has A as greatest element, so is a ρ-set by (ρ.1), that is, is in TρX. Thus Po ⊂ Tρ. This includes the submonad requirement prin(x) ∈ TρX. The other submonad requirement is precisely (ρ.6| ρ.6). TρX ⊂ BcX by definition. (2X, ⊃) → (22X, ⊃), A → 2A is order preserving, so maps ρ-sets to ρ-sets by (ρ.2). Thus {2A : A ∈ A}c is a ρ-set by (ρ.3), so Tρ satisfies (♣). The proofs for Tρ and for (♣♣) are similar.

6 CONTINUOUS MONADS 38

Proposition For a continuous monad T, every algebra is, in part, a poset with non-empty infima. Proof Every algebra ξ : TX → X is a Po-algebra PoX − → TX

ξ

− − → X with non-empty infima

A = ξ(Prin(A)).

Remark Let S ⊂ T be continuous monads and let (X, ξ) be a T-

• algebra. Thus

SX → TX

ξ

− → X is an S-algebra. But both are the same poset.

6 CONTINUOUS MONADS 39

Proposition For a continuous monad T, each free algebra (TX, µX) is closed under non-empty intersections. Proof

Ai = µX{Prin({Ai)}}

= µX{B ⊂ TX : {Ai} ⊂ B} = {W ⊂ X : {Ai} ⊂ ✷W} = {W ⊂ X : W ∈ Ai for all i} =

Ai

6 CONTINUOUS MONADS 40

Now let’s bring in a conditional ρ and explore why the main theorem part 1 is true. Recall TρX = {A ∈ BcX : A is a ρ-set in (2X\{∅}, ⊃)}

• TρX = {A ∈ BcX : A is a ρ-set in (2X, ⊃)}
• j∈J
• k∈K(j)

xjk =

• f∈ K(j)
• j∈J

xj,fj (CDρ) We’ve shown that Tρ, Tρ are continuous monads. In (CDρ), we have observed that the infima are intersections. Axioms (ρ.5) give that the ρ-suprema are unions. But it is trivial to show that complete distributivity holds in a power set with intersections and unions. In short: TρX is a ρ-distributive ρ-poset. Similarly Tρ.

6 CONTINUOUS MONADS 41

So we want to understand why the Tρ-algebras are the ρ-continuous posets. We’ve already seen that TρX is a ρ-poset satisfying (CDρ). Given another such object (Y, ≤) and a function f : X → Y , define f # : TρX → (Y, ≤) by, f #(A) =

• A∈A
• x∈A

fx The map (2X\{∅}, ⊃) → (Y, ≤), A →

fA

is order-preserving and so, for each A ∈ TX, maps the ρ-set A to the ρ-set {∧fA : A ∈ A} by (ρ.2). This shows f # is well-defined. The hard work here is to show that f # preserves infima. This is quite like the proof in Alan Day’s paper. The resulting adjointness leads to the same monad we started with.

6 CONTINUOUS MONADS 42

To finish things we must show the Beck coequalizer condition. This would be routine if we know that for a surjective morphism q : (X, ≤) → (Y, ≤)

• f ρ-posets, that (Y, ≤) is ρ-continuous if (X, ≤) is.

Lemma We can “pull back” ρ-sets: for q : (X, ≤) → (Y, ≤) as above, if E is a ρ-set in Y then there exists a ρ-set D in X with fD = E.

• Proof. Use the L-splitting lemma Y

g

− → X

f

− → Y = idY , and set D = gE. Corollary Quotients preserve equations involving infima and ρ-suprema.

6 CONTINUOUS MONADS 43

Here’s a table of examples of the main theorem part 1. ρ Monad Tρ Monad Tρ ρa Bc

• Bc

ρb I Bc ρd Fo F ρdb N F ρc I Bc ρg Po P (1) ρb is bounded sets. IX = {A ∈ BcX :

A = ∅}

ρdb is directed and bounded sets. N is the neighborhood monad. ρc is (finitely) consistent sets. IX = {A ∈ BcX : A has FIP}. Note that ρdb = ρd ∩ ρb. In general, any intersection of conditionals is a conditional. Conditionals form a complete lattice with least element ρg and greatest element ρa.

6 CONTINUOUS MONADS 44

Compare the next result to the fact that the free complete lattice on three elements is a proper class. Corollary A finitely-generated ρ-continuous poset is finite. For example, the free continuous lattice on 3 elements has 7 elements.

6 CONTINUOUS MONADS 45

The next result is well known for continuous lattices (Lemma I.1.12 of the Compendium). Corollary Each ρ-continuous poset is ρ-meet continuous, that is, the law (∨xi) ∧ x = ∨(xi ∧ x) (MCρ) holds whenever {xi} is a ρ-set. Proof The law trivially holds in (TρX, ⊂) or ( TρX, ⊂) and such a law is preserved by quotients because we can pull back ρ-sets.

6 CONTINUOUS MONADS 46

For a ρ-continuous poset X,

• x ∈ X is ρ-compact if x ≪ρ x.
• In TX, Prin(A) is ρ-compact.
• X is ρ-algebraic if for each x ∈ X, x =

{y ≤ x : y is ρ-compact}.

• (TρX, ⊂) is ρ-algebraic since for A ∈ TρX,

A =

{Prin(A) : A ∈ A

• Pfn(X, Y ) is an algebraic N-algebra. Partial functions with finite

domain are compact.

• A Scott domain is an algebraic N-algebra.

Example For the completely-distributive lattice (2X, ⊂), if a ∈ A then {a} ≪ρ {a} ≪ρ A because A ⊂

Di ⇒

∃ i a ∈ Di This is algebraic since A is the union of its singletons.

6 CONTINUOUS MONADS 47

Main Theorem Part 2 Let T be a continuous monad. Then there exists a largest conditional ρ such that T =

  

Tρ if {∅} / ∈ T∅

• Tρ

if {∅} ∈ T∅ Proof idea. Say that A ⊂ (X, ≤) is a ρ-set if its familification ΥA = {↑a : a ∈ A}c is in TX. Let γ : 2X → TX map A to Prin(A). To show TρX ⊂ TX observe that for A ∈ TρX, γ#({2A : A ∈ A}) = A. Axiom (♣) is needed to prove that TX ⊂ TρX. Axiom (♣♣) is needed to prove (ρ.4). Example A ⊂ (X, ≤) is directed if and only if ΥA is a filter.

6 CONTINUOUS MONADS 48

Mutual cofinality lemma Let T be a continuous monad with largest conditional ρ with T = Tρ. Let A ⊂ (X, ≤) be a ρ-set and let B ⊂ X be such that A, B are mutually cofinal. Then B is a ρ-set. Proof: {↑a : a ∈ A}c = {↑b : b ∈ B}c. For the largest ρ, we can improve the definition of ρ-continuous: Corollary Let T be a continuous monad with largest conditional ρ with T = Tρ. Then in any ρ-continuous poset (X, ≤), for each x ∈ X, {y : y ≪ρ x} is a ρ-set.

7 THE TWO-ELEMENT ALGEBRA 49

7 The Two-Element Algebra In this section we consider arbitrary (not necessarily continuous) family monads S ⊂ T ⊂ B. The forgetful functor SetT → SetS preserves limits. The complete atomic Boolean algebra 2 = {0, 1} with 0 < 1 restricts to a T-algebra χ{1} : T2 → B2

γ

− − → 2 We call this important T-algebra 2. Thus the product 2I of complete atomic Boolean algebras maps to the product 2I in SetT. If Prin{0, 1} ∈ T2, then 0 ∧ 1 = γ( Prin{0, 1}) = 0, so 0 < 1 in 2.

7 THE TWO-ELEMENT ALGEBRA 50

Now, if Po ⊂ T, a T-algebra (X, ξ) has non-empty intersections

A =

ξ(Prin(A)). This is the same operation for all the monads, including B. Thus 2I is the same poset (2I, ⊂) for all the monads T. Open question: Is there a submonad Po ⊂ T ⊂ B and two T-algebras (X, ξ), (X, θ) wlth ξ = θ but with both algebras having the same poset structure?

7 THE TWO-ELEMENT ALGEBRA 51

For A ⊂ X let ✷A = {A ∈ TX : A ∈ A}. Lemma For every A ⊂ X, the characteristic function (TX, µX)

χ✷A

− − − − → 2 is a T-homomorphism.

• Proof. This holds when T = B since BX is the product 22X and the

Ath projection function 22X → 2 is just χ✷A. Now consider TTX TX

✲

µX TBX BBX

✲

iBX BX = 22X

✲

νX T2 B2

✲

i2 2

✲

γ

❄

Tχ✷A

❄

χ✷A

❄

Bχ✷A

❄

TiX

❄

iX (i monad map) (i natural) (homo)

7 THE TWO-ELEMENT ALGEBRA 52

Corollary In SetT, TX is a subalgebra of 22X. Proof. TX 22X

✲

iX χ✷A

❅ ❅ ❅ ❅ ❅ ❘

2

❄

πA Thus 2 generates SetT as a variety, SetT = QSP(2).

7 THE TWO-ELEMENT ALGEBRA 53

Now consider, once again, the the following fundamental diagram for a T-algebra (X, ξ). X TX 22X

✲ ❄

ξ There are two non-trivial topologies on the ordinal 2 = {0, 1}, the Sierpi´ nski topology ({1} is open, {0} not) and the discrete topology. In each case, give TX the subspace topology of the product topology and then X gets the quotient topology. Respectively call these topolo- gies the Sierpi´ nski topology T´ n and the canonical topology Tcan. Observation The canonical topology contains the Sierpi´ nsky topology because this is true for 2.

7 THE TWO-ELEMENT ALGEBRA 54

Lemma Every T-homomorphism is continuous in both the Sierpi´ nsky topology and the canonical topology. Proof idea: Do the same in both cases:

• Show Bf is continuous for any function f.
• Then show Tf : TX → TY is continuous if TX, TY have the

subspace topology.

• Then use that ξ is a quotient map in Top.

TY BY

✲

iY TX BX

✲

iX

❄

Tf

❄

Bf Y

✲

πA πf −1A

❅ ❅ ❅ ❅ ❘

X Y

✲

f TX TY

✲

Tf

❄

ξ

❄

θ

7 THE TWO-ELEMENT ALGEBRA 55

Lemma For both topologies, TX is a subspace of 22X. Proof idea: Give TX and TTX their subspace topologies and show µX : TTX → TX is a quotient map as follows.

• BTX

BiX

− − → BBX

νX

− − → BX followed by a projection is a projec- tion.

• µX is continuous from the diagram below.
• We saw TηX : TX → TTX is continuous. Thus µX is a split epic

in Top hence a quotient. TTX TX

✲

µX BTX BBX

✲

BiX BX = 22X

✲

νX

❄

iTX

❄

iX (i monad map) Corollary In the canonical topology, TX is a completely regular zero- dimensional space.

7 THE TWO-ELEMENT ALGEBRA 56

Lemma Let S ⊂ T ⊂ B. If SX, TX have either the Sierpi´ nski or the canonical topology then SX is a subspace of TX. Proof idea. In Top, given inclusions i = X

j

− → Y

k

− → Z, if i and k are subspaces, so is j. This is because if z : D → E is a subspace and f is a function from a space into D, f is continuous if and only if zf is.

7 THE TWO-ELEMENT ALGEBRA 57

Lemma For any continuous monad, all Sierpi´ nski open sets of an al- gebra are upper sets.

• Proof. ✷(A1, . . . , Am) = {A ∈ TX : each Ai ∈ A} is a basic open

set in TX and this is clearly an upper set; any union of upper sets is

• upper. Thus it’s true in TX.

For (X, ξ) we have our L-splitting lemma X

g

− → TX

ξ

− → X = id so U ⊂ X open ⇒ ξ−1U upper If u ∈ U, u ≤ v then gu ≤ gv with gu ∈ ξ−1U so v = ξgv ∈ U.

7 THE TWO-ELEMENT ALGEBRA 58

For the algebra 2, the Sierpi´ nski topology is the Sierpi´ nski space and the canonical topology is the discrete space. Theorem For any continuous monad T with algebra (X, ξ) with the Sierpi´ nski topology, x ≤ y ⇔ x ∈ {y} Proof We must show {y} = ↓y. ξ−1(↓y) = {A : ξA ≤ y} = {A :

• A∈A

A ≤ x{

= {A : ∀A ∈ A,

A ≤ x}

Is this closed? Let Ai be a net in ξ−1(↓y), Ai ⇁ A ∈ TX. For A ∈ A, A is eventually in Ai so

A ≤ x.

We have ↓y is closed because its inverse image is. Since open sets are upper, closed sets are lower. Thus ↓y is the smallest closed set containing y, as needed.

7 THE TWO-ELEMENT ALGEBRA 59

Note however: The topologies for the algebra 2I vary with T and need not be the product topology. For T = Po, all upper sets are open so 2I has the box topology. For a T-algebra, if S ⊂ T then the S-topology contains the T-topology for both Sierpi´ nski and canonical.

7 THE TWO-ELEMENT ALGEBRA 60

An Alexandroff space is a topological space in which every intersec- tion of open sets is open. Exercise Use the L-splitting lemma to prove that the injective T0 Alexandroff spaces are the P-algebras. [Hint: PX is a power of Sierpi´ nski 2].

7 THE TWO-ELEMENT ALGEBRA 61

It is well known that βX = {U ∈ BcX : U is an ultrafilter} is a sub- monad of F. An algebra ξ : βX → X is a compact Hausdorff space with ξ its ultrafilter convergence function. Lemma The canonical topology of βX is the topology of the beta- compactification of discrete X. Proof A basic open set U in 22X is obtained by choosing A1, . . . , Am ⊂ X and B1, . . . , Bn ⊂ X and setting U = {A ∈ 22X : A1, . . . , Am ∈ A and B1, . . . , Bn / ∈ A} Standard properties of an ultrafilter reduce this to U = ✷A = {U ∈ βX : A ∈ U} which is a standard definition of a base for βX.

7 THE TWO-ELEMENT ALGEBRA 62

Theorem In the canonical topology, TX is compact for all X ⇔ βX ⊂ TX for all X. Proof. ⇒. X is dense in βX so, if U ∈ βX there exists a net prin(xi) which converges to U. Thus the net converges to U in 22X. But prin(xi) ∈ TX and TX is closed so U ∈ TX. ⇐. Let Φ : SetT → Setβ be the forgetful functor. Φ(2) is the two- element discrete space, the only Hausdorff topology on 2. As Φ pre- serves products, it maps the T-algebra 22X to the Cantor space. Thus φ maps the T-subalgebra i : TX → 22X to a continuous inclusion of TX into the Cantor space in the category of compact Hausdorff spaces, and any such inclusion is a subspace so that the compact topology on TX is the canonical topology.

7 THE TWO-ELEMENT ALGEBRA 63

The next example illustrates how Wyler [22] developed the Lawson topology. Example If T = F (filters) then clearly β ⊂ F. Thus each FX is compact and then each algebra (= continuous lattice) (X, ξ) is compact, being a quotient of FX. Here, the canonical topology coincides with what is usually called the Lawson topology. Thus once free algebras are compact, β ⊂ T so all algebras are compact Hausdorff. Example If T = N, no nonprincipal ultrafilter belongs to NX. Thus the canonical topology is not compact.

7 THE TWO-ELEMENT ALGEBRA 64

Lemma 2X ∈ TρX is an isolated point in the canonical topology.

• Proof. ✷A is open in the canonical topology and {2X} = ✷∅.

Corollary For the proper filter monad Fo, FoX is compact in the Lawson topology.

8 SCOTT MONADS 65

8 Scott Monads Let T be a continuous monad with largest conditional ρ. T is a Scott monad if (σ.1) T ⊂ F, that is, every ρ-set is directed. Examples F, Fo ane N are Scott monads.

8 SCOTT MONADS 66

In Harmonic analysis, one studies representations of Banach algebras. For any topological space, its space C(X) of bounded, continuous real- valued functions is a Banach algebra. As C(X) = C(βX), X might as well be compact Hausdorff. Gelfand showed that C(X) characterized

• X. Even more:

Theorem (Gelfand and Kolmogoroff [9]) For compact Hausdorff X, C(X) as a ring characterizes X.

• Proof. Consider the topological product R

RC(X) and let H be the sub- space of all ring homomorphisms. The map x → evx embeds X home-

• morphically onto H.

8 SCOTT MONADS 67

Gillman and Jerison [10] introduced P spaces as those topological spaces in which every countable intersection of open sets is open. They proved Theorem For a completely regular Hausdorff space X with ring of continuous real-valued functions C(X), the following are equivalent.

• 1. X is a P space.
• 2. Each prime ideal in C(X) is maximal.
• 3. For f, g ∈ C(X), (f, g) = (f 2 + g2).
• 4. Every finitely generated ideal in C(X) has an idempotent.

8 SCOTT MONADS 68

Such results to not apply to our Sierpi´ nski spaces since these are T0 but not T1. Let a ρ-set D be ω-directed, that is, for all sequences (dn) in D there exists d ∈ D with all dn ≤ d. Theorem ρ is the largest conditional for the Scott monad FωX = {F ∈ FX : F is closed under countable intersections} and each ρ-continuous poset in the Sierpi´ nski topology is a T0 P space.

• Proof. Use condition (1) of the next theorem.

8 SCOTT MONADS 69

If (X, ξ) is a T-algebra, we define a family Tξ as follows: Tξ = {U ⊂ X : F ∈ TX and ξ(F) ∈ U ⇒ U ∈ F} Theorem Let T be continuous. For U ⊂ X, (1) ⇔ (2) ⇒ (3). If T is Scott, then (3) ⇒ (1) (so in particular (1)and (2) are topologies).

• 1. U ∈ Tξ
• 2. U is an upper set and, for all ρ-sets D,

D ∈ U ⇒ U ∩ D = ∅.

• 3. U is open in the Sierpi´

nski topology. Open question: Are the open sets of (1), (2) always a subbase for the Sierpi´ nski topology?

8 SCOTT MONADS 70

From this point forward we add the following axiom to a Scott monad: (σ.2) If (Ai : i ∈ I) is a net in TX with

• i
• j≥i Aj ∈ TX then

{

• j≥i Aj : i ∈ I} is a ρ-set in (TX, ⊂).

Examples: F, Fo, N but not Fω.

8 SCOTT MONADS 71

Proposition A ⊂ X is a subalgebra of (X, ξ) if and only if A is closed under non-empty infima (or, all infima if 2X ∈ TX) and ρ-suprema.

• Proof. From the proof of the main theorem, part 1, the structure map

is ξ(A) =

• A∈A

A

so ξ(Prin(A)) =

A

Let R be a ρ-set so that {↑r : r ∈ R}c ∈ TX. Then ξ({↑r : r ∈ R}c) =

• ↑r ⊂D

D

=

• ↑r

↑r = R

8 SCOTT MONADS 72

Theorem Let ρ be a conditional and let Q ⊂ TρX be a subalgebra. Then Q is closed in the canonical topology. Proof. Let Ai be a net in Q which converges to A ∈ TρX. Then Ai converges to A in 22X so that for A ⊂ X, A ∈ A ⇔ A is eventually in Ai ⇔ ∃ i A ∈

• j≥i

Aj ⇔ A ∈

• i
• j≥i

Aj By axiom (σ.2) the union is a ρ-supremum, so as Q is a subalgebra, A ∈ Q.

8 SCOTT MONADS 73

Corollary A subalgebra of a ρ-continuous poset is closed in the canon- ical topology. Proof. For ξ : TρX → X, if Q is a subalgebra of X, ξ−1(Q) is a subalgebra of TX, hence is closed by the theorem. For a continuous lattice, ↑x = {y : y ≥ x} is not closed in the Scott

• topology. The Lawson topology is precisely that enlargement of the

Scott topology by requiring ↑x to be closed. Corollary If (X, ≤) is a ρ-continuous poset, ↑x is closed in the canon- ical topology for all x. Proof. ↑ x is closed under infima and non-empty suprema, so is a subalgebra.

8 SCOTT MONADS 74

Subalgebras need not be closed in the Sierpi´ nski topology. For example a singleton {a} is a subalgebra but is not a lower set hence is not Sierpi´ nski closed. This shows, by the way, that the canonical topology is always T1.

8 SCOTT MONADS 75

The next results are proved exactly as in the Compendium [8]. Proposition For an algebra (X, ξ), A ⊂ X is closed in Tξ ⇔ A is a lower set and for all ρ-sets B ⊂ A,

B ∈ A.

Proposition An algebra (X, ξ) satisfies the strong interpolation property that if x ≪ρ z with x = z then there exists x = y with x ≪ρ y ≪ρ z. Theorem For (X, ξ) an algebra, the sets of form xo = {y : x ≪ρ y} determine an open base for Tξ. Indeed xo is the interior of ↑x.

8 SCOTT MONADS 76

Theorem For (X, ξ) an algebra, the canonical topology is Hausdorff. Proof Let x ≤ y. x =

(u : u ≪ρ x) so there exists u ≪ρ x with

u ≤ y. Then x ∈ uo, y ∈ X\ ↑u are separating open sets. Proposition If N is the Tξ-neighborhood filter of x in (X, ξ), ξ(N) =

• U∈N

U = x

8 SCOTT MONADS 77

Nachbin’s book Topology and Order studies structures which are simul- taneously topological spaces and ordered sets. The following follows from the results above by arguments given early in the book. Proposition For an algebra (X, ξ), let X × X have the product topol-

• gy of the canonical topology. Then ≤ is closed in X × X.

REFERENCES 78

References [1] S. Abramsky and A. Jung, Domain theory, in S. Abramsky, D.

• M. Gubbay and T. S. E. Maibaum (eds.), Handbook of Logic in

Computer Science, vol. 3, Clarendon Press, Oxford, 1994. [2] P. S. Alexandroff, ¨ Uber die Metrisation der im kleinen kompakten topologische R¨ aume, Mathematische Annalen 92 (1924), 294–301. [3] P. S. Alexandroff and H. Hopf, “Topologie I”, Springer-Verlag, Berlin, 1935. [4] B. Banaschewski, The duality of σ-continuous lattices, Lecture Notes in Mathematics 871, Springer-Verlag, Berlin (1981), 12–19. [5] E. ˘ Cech, “Topological Spaces”, revised from the original 1959 Czech version by Z. Frolik and M. Kat˘ etov, Academia Publishing House of the Czechoslovak Academy of Sciences, Prague, 1966. [6] Day, A., Filter monads, continuous lattices and closure systems, Canadian Journal of Mathematics XXVII (1975), 50–59. [7] M. Ern´ e, A completion-invariant extension of the concept of a continuous lattice, Lecture Notes in Mathematics 871, Springer- Verlag, Berlin (1981), 45–60. [8] Giertz, G., K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove and D. S. Scott, “A compendium of continuous lattices”, Springer- Verlag, Berlin, Heidelberg and New York, 1980. [9] I. Gelfand and A. Kolmogoroff, On rings of continuous functions

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11–15. [10] L. Gillman and M. Jerison,“Rings of Continuous Functions”, D. van Nostrand, New York, 1960. [11] A. Grothendieck and colleagues SGA4.1, Th´ eorie des topos et co- homologie ´ etale des sch´ emes, Lecture Notes in Mathematics 269, Springer-Verlag, 1972.

REFERENCES 79

[12] C. Gunter, “Semantics of Programming Languages: Structures and Techniques”, MIT Press, 1992. [13] R. E. Hoffmann, Projective sober spaces, Lecture Notes in Mathe- matics 871, Springer-Verlag, Berlin (1981), 125–158. [14] P. T. Johnstone, “Stone Spaces”, Cambridge University Press, Cambridge (1982). [15] Manes, E. G., A class of fuzzy theories, Journal of Mathematical Analysis and its Applications 85 (1982), 409–451. [16] Manes, E. G., Monads of sets, in M. Hazewinkel (ed.), Handbook of Algebra, Vol. 3, Elsevier Science B.V., Amsterdam, 2003, 67–153. [17] G. Markowski, Free completely distributive lattices, Proceedings of the American Mathematical Society 76 (1979), 227–228. [18] E. Nelson, z-continuous algebras, Lecture Notes in Mathematics 871, Springer-Verlag, Berlin (1981), 315–334. [19] D. S. Scott, Continuous lattices, Lecture Notes in Mathematics 274, Springer-Verlag, 1972, 97–136. [20] G. Raney, Completely distributive lattices, Proceedings of the American Mathematical Society 3, Number 5 (1952), 677–680. [21] O. Wyler, Dedekind complete posets and Scott topologies, Lecture Notes in Mathematics 871, Springer-Verlag, Berlin (1981), 384– 389. [22] Wyler, O., Algebraic theories of continuous lattices, Lecture Notes in Mathematics 871, Springer-Verlag, Berlin (1981), 390–413.

9 SURVIVAL 80

9 Survival

Once again, those still awake have survived another Manes tutorial!