Priestley Sketchpad tutorial

Five short lessons on drawing in the Priestley Sketchpad. Each lesson has a figure you can work in: press Edit to get the tools inside the figure, Reset to start over, and Done to go back to viewing it; Sketchpad ↗ continues in the full sketchpad. If Priestley duality is new to you, open the Background box at the end of each lesson.

  1. Reading a diagram
  2. Points and relations
  3. Convergent sequences
  4. Closing the order
  5. βℕ and its remainder

1Reading a diagram

A diagram shows a Priestley space: its points and its order. Larger points sit higher, and only covering relations are drawn, so an edge from x up to y means x < y with nothing in between.

  • Here ⊥ lies below every xn, and every xn lies below ⊤.
  • The xn converge to x∞. The first few terms are drawn; the dots and the fading edges stand for the rest.
  • A filled point is isolated. A hollow point, like x∞, is a limit of other points.
  • Press Edit, then click a point to shade what lies above it (green) and below it (orange). Try ⊤, then x∞.
  • In the editor, the panel beside the drawing shows the checks (is the order a partial order, is it closed, is the space a Priestley space) and the properties of the space. Hover any of them for an explanation.
An antichain xn converging to x∞, with a bottom ⊥ and a top ⊤ added.
Background: Priestley duality

A Priestley space is a compact space with a partial order such that whenever x ≰ y, some clopen up-set contains x but not y. Its clopen up-sets form a bounded distributive lattice, and every bounded distributive lattice L arises in this way, from the space of prime filters of L ordered by inclusion. So a diagram of a space is also a picture of a lattice: the lattice of its clopen up-sets.

In the figure, the clopen up-sets are ∅, the whole space, and the sets {⊤} ∪ A where A is a finite set of xn or a cofinite set of xn together with x∞. They form the finite–cofinite algebra on ℕ with a new bottom and a new top.

2Points and relations

Press Edit in the figure, then:

  1. Press P (Point) and click on the sheet to place a point. Points snap to the grid; hold Alt to place freely.
  2. Press R (Relation), click a smaller element, then a larger one. Put your new point above ⊤.
  3. The relations you enter generate the order: the sketchpad adds everything that follows by transitivity and draws only the covers. Add ⊥ ≤ ⊤ and nothing new is drawn, since it already follows. The entered relations are listed under Generating relations in the panel.
  4. Press V (Select) to click and drag. Delete removes the selection, Ctrl+Z undoes.
  5. Drag b below ⊥. The edge from ⊥ to b now runs downward: it is drawn dashed, and the panel warns of one edge downward, since a Hasse diagram puts larger points higher. Undo.
  6. Now enter a ≤ ⊥: with R, click a, then ⊥. Since also ⊥ ≤ a, the two lie below each other: they are ringed in red, and the check Partial order fails (hover it to see why). A Hasse diagram cannot show such a cycle, so the edges through a and ⊥ are no longer drawn as before. Undo.
A diamond: ⊥ below a and b, both below ⊤.
Background: finite posets

Every finite poset is a Priestley space with the discrete topology. All of its subsets are clopen, so its clopen up-sets are just its up-sets, and every finite distributive lattice is the lattice of up-sets of a finite poset (Birkhoff).

The diamond has six up-sets: ∅, {⊤}, {a, ⊤}, {b, ⊤}, {a, b, ⊤} and the whole set. Ordered by inclusion they form a square with a new bottom and a new top.

3Convergent sequences

Infinite spaces are drawn from sequences. Press Edit in the figure, then:

  1. Select x (V, then click it) and press N (Repeat ×ℕ).
  2. Move the pointer: to the side for an antichain, up for an increasing chain x0 < x1 < …, down for a decreasing one. Click to place the copies; Enter places them to the right.
  3. The copies have no limit yet, so the check Compact fails. With the sequence selected, the panel offers two compactifications: One point adds a limit x∞, and Stone–Čech adds a remainder (Lesson 5). Choose One point.
  4. Drag a term to move just that term. Double-click a term to select the whole sequence and drag it all.
  5. Relations repeat termwise. Start again (Reset), place a point p above x and relate x ≤ p, then repeat x: every xn lies below p. After you add the limit, x∞ lies below p as well, which is the subject of Lesson 4.
A single point x, to be repeated into a sequence.
Background: the one-point compactification

In a sequence with one limit point, the terms are isolated and every neighbourhood of x∞ contains all but finitely many terms. The clopen sets are the finite sets of terms and their complements, which contain x∞.

For the increasing chain 0 < 1 < 2 < … converging to ω, the clopen up-sets are ∅ and the final segments [n, ω], so its lattice is the chain ω + 1 turned upside down. See the example ω + 1.

4Closing the order

The order of a Priestley space is closed: if xn ≤ p for every n and xn → x∞, then x∞ ≤ p. The sketchpad adds such relations by itself. In the figure only xn ≤ p was entered, yet x∞ lies below p.

  1. Press Edit and hover the check Closed: it lists the relations that closedness added.
  2. Open Settings and turn off Close the order. Now the check fails instead, and lists what a closed order would also need. Turn it back on.
  3. Relations between sequences close up too: if xn ≤ yn for each n, then x∞ ≤ y∞. See the example Fence converging to an edge.
Every xn lies below p; closedness adds x∞ ≤ p.
Background: why the order is closed

If x ≰ y in a Priestley space, a clopen up-set U contains x but not y. Then U × (X ∖ U) is an open neighbourhood of (x, y) containing no pair x′ ≤ y′, since an up-set containing x′ contains y′. So the order is a closed subset of X × X.

With infinitely many points you enter only a few generating relations; the sketchpad completes them by transitivity and by closedness at the limit points.

5βℕ and its remainder

The other compactification of a sequence is the Stone–Čech compactification βℕ. Its new points form the remainder ℕ*, drawn as one shaded region.

  1. The first figure starts with a sequence x0, x1, … that has no limit yet, as Repeat ×ℕ leaves it. Press Edit, click a term to select the sequence, and choose Stone–Čech in the panel. The terms become the points of ℕ, and the region, labelled x*, is the remainder.
  2. The region stands for all of ℕ* at once. With P place a point p above it, then with R click the region, then p: every point of ℕ* now lies below p, but no term does. Clicking a term instead relates just that term, and Shift-clicking a term relates the whole sequence.
A sequence x0, x1, … as Repeat ×ℕ leaves it, to be compactified.

In the second figure, dn ≤ xn for each n, and xn → x∞. Every point of ℕ* is a limit of the dn, so closedness puts all of ℕ* below x∞. Press Edit and hover Closed to see it.

dn ≤ xn termwise, with xn → x∞; closedness forces ℕ* ≤ x∞.
Background: βℕ

βℕ is the Stone space of the power-set algebra 𝒫(ℕ). Its points are the ultrafilters on ℕ: the principal ones are the points n, and the free ones form the remainder ℕ* = βℕ ∖ ℕ. The clopen sets are exactly the closures of subsets of ℕ.

ℕ* has 2𝔠 points and none of them is isolated, so the sketchpad draws it as one region, and a relation with the region holds for every point of ℕ*. With the trivial order, βℕ is the dual of 𝒫(ℕ); with ⊥ and ⊤ added, as in the example βℕ between ⊥ and ⊤, its lattice is 𝒫(ℕ) with a new bottom and a new top.

Next

Further lessons will cover sequences of βℕ copies, space blocks, the properties panel, turning a diagram upside down, and finding and sharing examples. Meanwhile, the Examples window in the sketchpad lists every example, and can filter them by the properties of the space.