Relation algebras

Robin Hirsch and Ian Hodkinson

Thanks to the organisers for inviting us!And Happy New Year!

Workshop outline

1. Introduction to relation algebras

2. Games

3. Monk algebras: completions, canonicity

4. Rainbow construction: non-finite axiomatisability andnon-elementary results

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This talk: introduction to relation algebras

1. Algebras of relations

• boolean algebras, relation algebras, representations

• applications

2. Duality

• atom structures, representations of atomic relation algebras

• examples: point algebra, McKenzie & anti-Monk algebras

• ultrafilters, canonical extensions

• completions

3. Conclusion; some references

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1. Algebras of relations

Boole: started the algebraic formalisation of unary relations.

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Boolean algebras

– algebras (B,+,−, 0, 1) satisfying these equations, for all a, b, c ∈ B:

1. (a + b) + c = a + (b + c)

2. a + b = b + a

3. a + a = a

4. −(−a) = a

5. a + (−a) = 1

6. −1 = 0

7. a · (b + c) = a · b + a · c, where a · b abbreviates −(−a + −b)

8. 0 + a = a

We let a ≤ b abbreviate a + b = b, and a < b abbreviate a ≤ b ∧ a �= b.For S ⊆ B,

∑S is least upper bound of S in B, if exists.

∏S: glb.

We sometimes use − as a binary operator: a − b = a · (−b).

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Boolean algebras and unary relations

Definition: for any set X,

• a unary relation on X is just a subset of X,

• the algebra of all unary relations on X is (℘(X),∪, , ∅, X).The operations are the ‘natural’ ones.

• an algebra of unary relations on X is any subalgebra of this.Such algebras are also known as fields of sets.

These algebras are boolean algebras (exercise).

Conversely, any boolean algebra is isomorphic to an algebra ofunary relations on some set (Stone, 1936).

So boolean algebra axioms are sound and complete for unaryrelations: every boolean algebra is isomorphic to a field of sets.

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De Morgan

(born 27 June 1806 in Madura, Madras Presidency — now Madurai,Tamil Nadu)

— should consider binary (and higher-arity) relations.

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Binary relations?

• A binary relation on a set X is a subset of X × X.

• Egs: graphs, orderings, equivalence relations. Very important.

• An algebra of binary relations on X is a subalgebra of

Re(X) = (℘(X × X), ∪, , ∅, X × X, IdX , −−1, | ),

where

IdX = {(x, x) : x ∈ X},R−1 = {(y, x) : (x, y) ∈ R},R |S = {(x, y) : ∃z((x, z) ∈ R ∧ (z, y) ∈ S)}.

This choice of relational operations can be disputed.It does not lead to such a nice picture as for boolean algebras.

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Example: family relations

Let X be the set of all people (alive or dead).

Consider the binary relations son, daughter on X:(x, y) ∈ son iff y is a son of x, etc. Then

• child = son ∪ daughter

• grandson = child | son

• granddaughter = child | daughter

• parent = child−1

• sister = (parent | daughter) ∩ IdX

• aunt = parent | sister

• mother = parent | ((parent | daughter) ∩ IdX )

Exercise: try to do sibling, niece, cousin.

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Next developments

Peirce and Schroder established many properties of binary relations.

But no end in sight. . .

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C. S. Peirce

The logic of relatives is highly multiform; it ischaracterized by innumerable immediate conclusions fromthe same set of premises. . . . The effect of thesepeculiarities is that this algebra cannot be subjected to hardand fast rules like those of the Boolian calculus; and all thatcan be done in this place is to give a general idea of the wayof working with it.

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Tarski

— tried to reformulate Schroder’s results with modern algebra.

He wanted to axiomatise the algebras of binary relations.

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Relation algebras

In 1940s, Tarski proposed axioms.

They define the class RA of ‘relation algebras’:algebras A = (A,+,−, 0, 1, 1

,,˘, ; ) such that

• (A,+,−, 0, 1) is a boolean algebra

• (A, ; , 1,) is a monoid (a semigroup with identity, 1

,)

• Peircean law: (a ; b) · c �= 0 ⇐⇒ (a ; c) · b �= 0 ⇐⇒ a · (c ; b) �= 0for all a, b, c ∈ A.

c

ba

���

�����

���

Remark: Tarski’s original axioms were equations. They capture alltrue equations about relations that can be proved with 4 variables.

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Representable relation algebras — RRA

An algebra A = (A,+,−, 0, 1, 1,,˘, ; ) is said to be representable if it

is isomorphic to a subalgebra of a product of algebras of the formRe(X).

That is, there is an embedding

h : A →∏i∈I

Re(Xi),

for some sets I and Xi (i ∈ I).

Such an embedding is called a representation of A.The base set of h is

⋃i∈IXi (the Xi can be assumed disjoint).

The class of representable algebras is denoted RRA.RRA stands for ‘representable relation algebras’. Easy: RRA ⊆ RA.RRA is our main object of study in these talks.

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Why products?

An algebra of relations is a subalgebra of some Re(X).

So why are the representable algebras defined as those isomorphicto subalgebras of products of Re(X)s?

Likely answer: Tarski wanted the representable algebras to form avariety (equationally axiomatised class). So we get a closer notion toboolean algebras (also defined by equations).

Varieties are closed under products. Tarski proved (1955) that RRA,defined with products as above, is a variety.

Remark: a relation algebra A is simple if it has no nontrivial properhomomorphic images. This holds iff A |= ∀x(x > 0 → 1 ;x ; 1 = 1).

Any simple A ∈ RRA has a representation h : A → Re(X).

So for simple relation algebras, we can relax about products.

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Did Tarski’s axioms capture RRA?

Soundness (RRA ⊆ RA) is easy.

Completeness fails.

Lyndon (1950): gave example of A ∈ RA \ RRA. (|A| = 256)

Monk (1964): RRA is not finitely axiomatisable.

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More facts about RRA

Many ‘negative’ results about RRA are now known.

• RRA cannot be axiomatised by any set of equations using finitelymany variables (Jonsson 1988, but Tarski knew in 1975)

• Andreka has results on numbers of occurrences of operations inaxioms for representable cylindric algebras (analogous to RRA)

• there is no algorithm to decide whether a finite relation algebra isrepresentable (Hirsch–IH 1999)

• RRA is not closed under (Monk) completions (IH 1997)

• there is no Sahlqvist or even canonical axiomatisation of RRA(IH–Venema 1997, 2003)

Others for related classes coming later. . .

Compare the situation for boolean algebras. . . !

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Problems

1. Find simple intrinsic characterisation of (algebras in) RRA.

The next talk (games) contributes to this.

2. Finitization problem: find expressive operations on binary (andhigher-arity) relations, yielding a finitely axiomatisable class ofrepresentable algebras.

This is open. Sain, Sayed Ahmed and others have madeprogress on infinite-arity relations (mainly without equality).

A positive solution could contribute to a finitely axiomatisablealgebraisation of first-order logic. The Boolian paradise would beregained.

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Applications of relation algebras

Binary relations are fundamental. Results about them, and proofmethods, will have applications elsewhere.

1. Artificial planning: Allen, Ladkin, Maddux, Hirsch

2. Databases: use similar relational operations

3. Modal logic: product logics between Kn and S5n (n ≥ 3) areundecidable, non-finitely axiomatisable, no algorithm to decide ifa finite frame validates the logic (Hirsch–IH–Kurucz 2002)

4. Temporal logic: 1-variable first-order CTL∗ is undecidable(IH–Wolter–Zakharyaschev 2002)

5. Temporal logic: interval logics with Chop and the like are notfinitely axiomatisable (IH–Montanari–Sciavicco 2007)

6. Proof methods contributed to solution of Fine’s canonicityproblem (Goldblatt–IH–Venema 2003).

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2. Duality for relation algebras

Our aim in these talks is to study relation algebras and sketch proofsof some key results.

Duality is often helpful.

It is like taking logs in arithmetic — makes life easier.

But the main problems involved in finding representations of relationalgebras are hardly touched by duality.

In fact relation algebras shed as much light on duality as vice versa!

We will look at

• atoms, atomic relation algebras, atom structures

• ultrafilters, complete representations, canonical extensions

• completions

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2.1 Duality for algebras, by atoms

A relation algebra is a boolean algebra with extra operations. So wecan define a · b, a ≤ b,

∑S, etc., as in boolean algebras.

An atom of a relation algebra (or boolean algebra) is a ≤-minimalnon-zero element x: it satisfies ∀y(y < x ↔ y = 0).

A relation algebra A = (A,+,−, 0, 1, 1,, , ;) is atomic if every

non-zero element of A has an atom beneath (≤) it.

• For any X, Re(X) is atomic.• Any finite relation algebra is atomic (exercise).• There are infinite atomless relation algebras.

In an atomic relation algebra A, every element a of A is the sum ofthe atoms beneath it:

a =∑

{x : x an atom of A, x ≤ a}.

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Completely additive functions

Let (B,+,−, 0, 1) be a boolean algebra.

A function f : Bn → B is completely additive if for every i ≤ n,b1, . . . , bi−1, bi+1, . . . , bn ∈ B, and S ⊆ B such that

∑S exists,

f(b1, . . . , bi−1,∑

S, bi+1, . . . , bn)

=∑

{f(b1, . . . , bi−1, s, bi+1, . . . , bn) : s ∈ S}.

The RA axioms show that ˘ and ; are completely additive: if ai, bj areelements of any relation algebra, then whenever the blue sums exist,

(∑

i

ai)� =∑

i

ai

(∑

i

ai) ; (∑

j

bj) =∑i,j

(ai ; bj).

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Atomic relation algebras and atom structures

We can now specify atomic relation algebras quite easily.

We saw that ˘ and ; are completely additive. So in an atomic relationalgebra A, they are determined by their values on atoms.

(Every element of A is the sum of the atoms beneath it. So bycomplete additivity, a ; b =

∑{x ; y : x, y atoms, x ≤ a, y ≤ b}.)

So given its boolean part, A is determined by its atom structure AtA:• the set of atoms of A,• which atoms are ≤ 1

,,

• x, for each atom x (turns out that x is also an atom),• for each atoms x, y, z, whether x ; y ≥ z. (This determines x ; y.)

If x ; y ≥ z, we say that (x, y, z) is a consistent triple of atoms.Remark: (x, y, z) is consistent iff its Peircean transforms(x, y, z), (x, z, y), (z, y, x), (y, z, x), (z, x, y), (y, x, z) are all consistent.

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Examples: two finite relation algebras

1. McKenzie’s algebra K.

4 atoms: 1,, <, >, �.

1,= 1

,, < = >, > = <, � = �.

All triples are consistent except Peircean transforms of:(1

,, a, a′) for a �= a′, (<, <, >), (<, <, �), (�, �, �).

2. The ‘anti-Monk algebra’ M (Maddux?)

4 atoms: 1,, r, b, g.

a = a for all atoms a (‘symmetric algebra’).

All triples are consistent except Peircean transforms of:(1

,, a, a′) for a �= a′, and (r, b, g).

These are both relation algebras.Can you tell if they are in RRA or not?

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Relation algebra atom structures, complex algebras

Abstractly, a relation algebra atom structure is a structureS = (S, Id, ˘, C) satisfying certain first-order conditions (the‘correspondents’ of Tarski’s RA axioms).

If A is an atomic relation algebra then AtA is such a structure.

Conversely, given such an S, we can form its complex algebra:

S+ = (℘(S),∪, , ∅, S, 1,, ˘, ; ),

where• 1

,= {x ∈ S : S |= Id(x)}

• a = {x : x ∈ a} (for a ⊆ S)• a ; b = {z ∈ S : ∃x ∈ a ∃y ∈ b(S |= C(x, y, z))} (for a, b ⊆ S).

S+ is an atomic relation algebra, and AtS+ ∼= S.If A is an atomic relation algebra, A ↪→ (AtA)+.

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2.2 Duality for representations, by ultrafilters

Atomic RAs are determined by atoms. Are their representations?

Fix any (not necessarily atomic) relation algebra A, and arepresentation h : A →

∏i∈I Re(Xi), with base set X =

⋃Xi.

Each a ∈ A induces a binary relation h(a) on X. For x, y ∈ X let

h∗(x, y) = {a ∈ A : (x, y) ∈ h(a)}.

If (x, y) /∈ h(1) then h∗(x, y) = ∅. Otherwise, writing f = h∗(x, y):

1. 1 ∈ f

2. if a ∈ f and a ≤ b then b ∈ f

3. if a, b ∈ f then a · b ∈ f

4. for all a ∈ A, either a ∈ f or −a ∈ f (not both).

There is a name for such a subset of a boolean algebra: an ultrafilter.So for all x, y ∈ X, h∗(x, y) is either ∅ or an ultrafilter on A.

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Coherence conditions

The relational operations yield more properties of the h∗(x, y):

5. for any x, y ∈ X, we have 1, ∈ h∗(x, y) iff x = y.

6. for any x, y ∈ X, we have h∗(x, y) = {a : a ∈ h∗(y, x)}.

7. for any x, y, z ∈ X and a ∈ h∗(x, z), b ∈ h∗(z, y), c ∈ h∗(x, y),we have (a ; b) · c �= 0.

c

ba

���

�����

���x y

z

8. for any x, y ∈ X and a, b in A, if a ; b ∈ h∗(x, y),then there is some z ∈ X with a ∈ h∗(x, z) and b ∈ h∗(z, y).

9. For any non-zero a ∈ A, there are x, y ∈ X with a ∈ h∗(x, y).

These conditions are equivalent to h being a representation of A.This gives us a dual view of representations.

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Principal ultrafilters

An ultrafilter f of a relation algebra A is principal if it is of the form{a ∈ A : a ≥ x} for some x ∈ A.

In that case, x =∏

f , and x is an atom of A.

An ultrafilter is principal iff it contains an atom.

• Every ultrafilter of a finite relation algebra is principal.

• Every infinite relation algebra has non-principal ultrafilters.

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2.3 Bringing them together: representations of finite relation algebras

Let A be a finite (non-trivial simple) relation algebra.Fix a representation h : A → Re(X) of A.As each h∗(x, y) is principal, let the atom

∏h∗(x, y) stand for it.

So h can be viewed dually very simply, as a complete labelleddigraph M = (X, λ), where X is a set and λ : X × X → AtA is thelabelling. For all x, y, z ∈ X,

• λ(x, y) ≤ 1,

iff x = y.

• λ(x, y) = λ(y, x)�.

• λ(x, y) ≤ λ(x, z) ;λ(z, y). That is, ‘all triangles are consistent’.

• For all a, b ∈ AtA, if λ(x, y) ≤ a ; b then there is w ∈ X withλ(x, w) = a and λ(w, y) = b.‘All consistent triples are witnessed wherever possible.’

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2.4 Complete representations

Now let h be a representation of an infinite relation algebra A.The ultrafilters h∗(x, y) need not be principal — even if A is atomic.

Let’s examine the case when they are all principal.

• h is said to be an atomic representation if every ultrafilter h∗(x, y)is principal.

• h is said to be a complete representation if it preserves allexisting sums: for every S ⊆ A such that

∑S exists,

h(∑

S) =⋃{h(a) : a ∈ S}.

Fact (exercise):A representation of a relation algebra is atomic iff it is complete.

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Properties of complete representations

1. All representations of finite relation algebras are complete.

2. All infinite A ∈ RRA have incomplete representations.

3. Any completely representable relation algebra (one with acomplete representation) is atomic. (Converse fails.)

We can dually characterise any complete representation by adigraph with atoms labeling edges (as for finite relation algebras).

Let CRA be the class of completely representable relation algebras.

• CRA is properly contained in RRA.

• RRA is closure of CRA under subalgebras (see next slide).

• CRA is non-elementary (see Robin’s 2nd talk).

Problem: does CRA have the same first-order theory as the class offinite representable relation algebras?

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2.5 Canonical extensions (Jonsson–Tarski, 1951)

Let A be a relation algebra. The set of ultrafilters of A can be madeinto a relation algebra atom structure A+ (the canonical frame of A):

• f ≤ 1,

iff 1,A ∈ f

• f = {a : a ∈ f}

• f ; g ≥ h iff a ∈ f, b ∈ g ⇒ a ; b ∈ h.

The relation algebra Aσ := (A+)+ (complex algebra over theultrafilters) is called the canonical extension of A.

A embeds into Aσ via a �→ {f ∈ A+ : a ∈ f}, so we regard A ⊆ Aσ.

Fact (Monk, ∼1966): A ∈ RRA ⇐⇒ Aσ ∈ RRA.So RRA is what’s called a canonical variety.

Indeed, if A ∈ RRA then Aσ has a complete representation.

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2.6 (McNeille) completions (Monk 1970)

A completion of a relation algebra A is a relation algebra A such that

1. A ⊆ A,

2. A is complete as a boolean algebra:∑

S exists for all S ⊆ A,

3. A is dense in A: ∀c ∈ A \ {0} ∃a ∈ A \ {0} (a ≤ c).

Monk (1970) showed that completions of relation algebras alwaysexist and are unique up to isomorphism.

Easy fact: If A is an atomic relation algebra, then its completion is(AtA)+ — the complex algebra over its atom structure.

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Some facts about completions

The completion of a relation algebra A is somewhat analogous to itscanonical extension.They both extend A and are complete as boolean algebras.

However,

• Aσ is always atomic.A is atomic iff A is atomic.

• A preserves all existing sums and products of elements of A.For infinite A, Aσ never does (consider a non-principal ultrafilter).

• RRA is closed under canonical extensions (Monk).We show later that RRA is not closed under completions.

So canonical extensions of relation algebras seem the more useful.

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Conclusion

We have seen

1. boolean algebras, relation algebras (RA), representable relationalgebras (RRA)

2. simple relation algebras: have representations (if any) of formh : A → Re(X)

3. atomic relation algebras, atom structures, complex algebras

4. representations as graphs with edges labeled by ultrafilters (oratoms, for finite relation algebras)

5. complete representations — when the ultrafilters are principal

6. canonical extensions, completions

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Coming next

• Games

• Monk algebras, completions, canonicity

• rainbow algebras: RRA not finitely axiomatisable, CRAnon-elementary

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Some references

R. Hirsch, I. Hodkinson, Complete representations in algebraic logic,J. Symbolic Logic 62 (1997) 816–847.

R. Hirsch, I. Hodkinson, Representability is not decidable for finiterelation algebras, Trans. Amer. Math. Soc. 353 (2001) 1403-1425.

B. Jonsson, A. Tarski, Boolean algebras with operators I, Amer. J.Math. 73 (1951) 891–939.

J. D. Monk, On representable relation algebras, MichiganMathematics Journal 11 (1964) 207–210.

J. D. Monk, Completions of boolean algebras with operators,Mathematische Nachrichten 46 (1970), 47–55.

A. Tarski, On the calculus of relations, J. Symb. Logic 6 (1941) 73-89.

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