Operads in algebra, combinatorics, and computer science

Operads in algebra, combinatorics, and computer science

Samuele Giraudo

LIGM, Université Gustave Eiffel

January, February and March, 2021

Outline

1. Introduction 2. Collections 3. Treelike structures 4. Operads

5. Applications

Chapter

1. Introduction

Section

1.1 Algebraic structures

Types of algebraic structures

Combinatorics deals with sets (or spaces) of structured objects:

monoids;

groups;

lattices;

associative alg.;

Hopf bialg.;

Lie alg.;

pre-Lie alg.;

dendriform alg.;

duplicial alg.

Suchtypes of algebrasare specified by 1. a collection of operations;

2. a collection of relations between operations.

– Example –

The type of monoids can be specified by 1. the operations?(binary) and1(nullary);

2. the relations(x1 ? x2)? x3=x1 ?(x2 ? x3)andx ? 1=x=1? x.

Operator structures

– Strategy to study these algebras –

Add a level of indirection and consider algebraic structures wherein elements are operations.

There are a lot of kind of operator structures, each dealing with a particular type of operators:

x

1 2 3 4

Operads

x

1 2 3 4

a

b acc

Colored operads

x

2 3 1

Symmetric operads

x

1 2 3 4

1 2 3

Pros

x

1 3 1 2 2

Abstract clones

Operator structures and compositions

These operators can be composed in different ways.

Operads:

1 2 3

f ◦

"

1 2 g1 ,

1 g2,

1 2 g3

#

=

1 2 3 4 5

g1 f g2 g3

Pros:

f 1 2

1 2

∗ ^g

1 2 3 4

1 2 3

= f 1 2

1 2 g 3 4 5 6

3 4 5

f 1 2

1 2 3

◦ ^g

1 2 3

1 2 3 4

= ^f

1 2

g

1 2 3 4

Clones:

1 2 1

f

"

g1

1 3 2 3

,

1 3 2 g2

#

= g1 g1

f g2

Operator structures and application areas

Operadsare suitable to study algebraic structures where operations satisfy relations involving planar termslike

(x₁? x₂)•x₃=x₁•(x₂? x₃)−(x₁•x₂)•x₃.

Symmetric operadsare suitable to study algebraic structures where operations satisfy relations involvinglinear termslike

x₁? x₂=x₁•x₂+x₂? x₁.

Prosare suitable to study algebraic structures where operations can haveseveral inputs and outputsand satisfy relations involving linear terms like

∆ ((x₁? x₂)? x₃) = ∆ (x₁?(x₃? x₂)) + ∆ (x₁•(x₂? x₃)).

Clonesare suitable to study algebraic structures where operations can satisfyany relationlike x1? x1=x1+ (x2•x1)? x2.

Section

1.2 Computing over operations

Dendriform algebras

Adendriform algebra[Loday, 2001]is a spaceAendowed with two binary linear operations

≺,:A ⊗ A → A satisfying the three relations

(x₁≺x₂)≺x₃=x₁≺(x₂≺x₃) +x₁≺(x₂x₃), (x₁x₂)≺x₃=x₁(x₂≺x₃),

(x1≺x2)x3+ (x1x2)x3=x1(x2x3).

– Example –

OnKh{a,b}^∗i, let≺andbe the operations defined by

u≺v:=u^←v, uv:=u^→v, Then, for instance,

ab≺ba=abab+baab+baab, abba=abba+abba+baba.

(Kh{a,b}^∗i,≺,)is a dendriform algebra[Loday, 2001].

Dendriform associative operations

A binary operation?isassociativeif

?

? = ^? _? .

By using the infix notation, this says that

(x1? x2)? x3=x1?(x2? x3).

– Proposition –

In any dendriform algebra(A,≺,), any associative operation is proportional to the binary operation

≺+.

To prove this, let us consider a generic binary dendriform operation

t :=λ₁ ≺ +λ₂ ,

whereλ₁andλ₂are any coefficients ofK.

Dendriform associative operations

This operation is associative iff

t

t − ^t

t =λ²₁

≺

≺ +λ1λ2

≺

+λ1λ2

≺ +λ²₂

−λ²₁ ^≺

≺ −λ1λ2

≺

−λ1λ2

≺ −λ²₂

= 0.

By the three dendriform relations, this is equivalent to

λ²₁

≺

≺ −λ²₁ ^≺

≺ −λ1λ2

≺

= 0, λ²₁λ2

≺ +λ²₂

−λ²₂ = 0.

λ1λ2

≺

−λ1λ2

≺ = 0,

which is itself equivalent toλ²₁=λ₁λ₂=λ²₂. Therefore,λ₁=λ₂.

Duplicial algebras

Aduplicial algebra[Brouder, Frabetti, 2003]is a spaceAendowed with two binary linear operations ,:A ⊗ A → A

satisfying the three relations

(x1x2)x3=x1(x2x3), (x1x2)x3=x1(x2x3), (x1x2)x3=x1(x2x3).

– Example –

OnKhN^∗i, letandbe the operations defined by uv:=u v↑_max(u)

, uv:=u v↑_|u|

. Then, for instance,

021114 = 021136, 021114 = 021158.

(KhN^∗i,,)is a duplicial algebra[Novelli, Thibon, 2013].

Duplicial operations and equivalence

Two operationstandt⁰areequivalent(writtent≡t⁰) if they produce the same output when evaluated with the same inputs.

Let us describe as way to test if two duplicial operations are equivalent.

By duplicial relations, we have

≡

,

≡

,

≡

.

We orient them as

→

,

→

,

→

.

in order to obtain a rewrite relation⇒on the set of all the duplicial operations by performing local moves.

Testing equivalence of duplicial operations

We have for instance the sequence

⇒

⇒

of rewritings.

– Proposition –

Two duplicial operationstandt⁰are equivalent iff there is a duplicial operationssuch thatt⇒^∗ sand t⁰⇒^∗ s.

To prove this, we have to establish the fact that⇒is a terminating and confluent rewrite relation.

Enumerating duplicial operations

The set of all normal forms of⇒contains duplicial operations that are pairwise nonequivalent.

– Proposition –

The set of normal forms of⇒of operations withn>0inputs is in one-to-one correspondence with the set of all binary trees withninternal nodes.

A possible bijection puts the following two trees in correspondence:

←→ .

Therefore, there are

cat(n) := 1 n+ 1

2n n

pairwise nonequivalent duplicial operations withninputs.

Pre-Lie algebras

Apre-Lie algebra[Vinberg, 1963] [Gerstenhaber, 1963]is a spaceAendowed with a binary operation x:A ⊗ A → A

satisfying the relation

(x1xx2)xx3−x1x(x2xx3) = (x1xx3)xx2−x1x(x3xx2).

– Example –

OnKh{a,b}^∗i, letxbe the operation defined by

uxv:= X

16i6|u|−1

u1. . . uiv ui+1. . . u_|u|.

Then, for instance,

aababxbb=abbabab+aabbbab+aabbbab+aababbb

=abbabab+ 2aabbbab+aababbb.

(Kh{a,b}^∗i,x)is a pre-Lie algebra.

Description of pre-Lie operations

One can ask the same question as before concerning the description of equivalence classes of pre-Lie operations.

– Theorem

–

There is a one-to-one correspondence between the set of all pre-Lie operations withn>0inputs the set of all standard rooted trees withnnodes.

To prove this, one can consider a symmetric operadPLieon rooted trees and show that free algebras overPLieare free pre-Lie algebras.

Here is a composition of pre-Lie operations encoded by standard rooted trees:

1 3

2

◦3 ¹ 2

=

1 3 2 4

+ 4 2 3

1 +

4 1 3

2 + ³

1 4

2

, and here is a pre-Lie product in the free pre-Lie algebra over one generator:

x = + + + 2 .

Section

1.3 Computing over combinatorial objects

Objects as operators

Byregarding objects as operators, one obtains ways tocomposethem. For instance,

1 2

4 3

5

◦_{2 1}² =

1 1

2 4 3

5

=

1 2

3 5 4

6

is an abstract composition of an object of size5with an object of size2at the2-nd position.

Concrete examples on Motzkin paths:

◦4 = ,

on permutations:

•

••

•• ◦₃

•

•• =

•

•

•

••

••

,

and on some labeled graphs:

−1 2 1◦2 ⁻¹

1

1 = ⁻¹⁻¹

2 1

1

+ 2 ⁻¹

2 1

1

+ ⁻¹1 2

1

1

.

Enumeration of Motzkin paths

AMotzkin pathis a path inN²starting from(0,0)and ending at(n,0), made of steps(+1,+1), (+1,−1), and(+1,0).

– Example –

With the aid of some elementary reasoning, one can prove that the generating seriesG(t)of Motzkin paths, enumerating them w.r.t. their number of points, satisfies

G(t) =t+tG(t) +tG(t)² and

G(t) =t+t²+ 2t³+ 4t⁴+ 9t⁵+ 21t⁶+ 51t⁷+ 127t⁸+ 323t⁹+· · · .

Composition of Motzkin paths and series of objects

A way to obtain this series consists in following both steps:

1. define a composition operation on the set of Motzkin paths;

2. express the infinite formal sum of all Motzkin paths.

Ifuandvare two Motzkin paths, the compositionu◦ivis obtained by replacing thei-th point ofubyv.

The infinite formal sum of all Motzkin paths is

f := + + + + + + + +· · ·,

and one can prove that it satisfies the functional equation

f = + ◦[ ,f] + ◦[ ,f,f].

This is a consequence of a property of the operadMotzof Motzkin paths (and more precisely, the fact that it is a Koszul operad).

Section

1.4 Mains application fields and purposes

Purposes of the lecture

We focus in this lecture onoperads.

Operads are used to study algebraic structures. They are a tool to compute over operations;

describe the combinatorial heart of a type of algebraic structure;

relate, by transformations, different types of algebraic structures.

Conversely, operads can also be used as tools to interpret combinatorial objects as operations;

describe how to form objects from elementary building blocks;

enumerate families of combinatorial objects;

generate families of combinatorial objects.

Main topics

We shall consider and study

collections, that are structured sets of combinatorial objects;

treelike structureof several types (planar rooted trees and syntax trees);

rewrite systemson syntax trees and give ways to prove termination and confluence;

nonsymmetric operadson combinatorial objects, constructions, and study methods to prove presentations;

some applicationsand open questions.

Section

1.5 Exercises

About types of algebras

– Exercise –

LetA:={a₀,a1,a2, . . .}be an infinite alphabet totally ordered byai6ajiffi6j, and let≺andbe the two binary operations defined onK

A⁺ by

u≺v:=

(uv ifmax₆(u)>max₆(v) 0 otherwise,

uv:=

(uv ifmax₆(u)<max₆(v) 0 otherwise.

Prove that K A⁺

,≺,

is a dendriform algebra.

– Exercise –

Prove that(Kh{a,b}i,^←,^→)is a dendriform algebra.

– Exercise –

Let(A,≺,)be a dendriform algebra and setxas the binary operation defined byxxy:=x≺y−yx. Prove that(A,x)is a pre-Lie algebra.

About types of algebras

– Exercise –

Let(A,x)be a pre-Lie algebra and set[−,−]as the binary operation defined by[x, y] :=xxy−yxx. Prove that (A,[−,−])is a Lie algebra.

Recall that a Lie algebra is a spaceAendowed with a binary linear operation[−,−]satisfying [x, y] =−[y, x],

[x,[y, z]] + [y,[z, x]] + [z,[x, y]] = 0.

– Exercise –

Prove that each duplicial operation withn>0inputs can be encoded by a binary tree withninternal nodes.

– Exercise –

Express all associative operations in a duplicial algebra.

Chapter

2. Collections

Section

2.1 Collections and enumeration

Collections

Anindex setis a nonempty setI.

AnI-collectionis a setCdecomposing as a disjoint union C=G

i∈I

C(i)

where theC(i)are possibly infinite sets.

For anyx∈C(i),xis anobjectofCandind(x) :=iis theindexofx.

When allC(i)are finite,Ciscombinatorial.

LetC₁andC₂be twoI-collections.

A mapφ:C1→C2is anI-collection morphismif for allx∈C1, ind(x) = ind(φ(x)).

If for alli∈I,C₁(i)⊆C₂(i), thenC₁is asubcollectionofC₂.

Graded collections

Agraded collectionis anN-collection.

LetCbe a graded collection. For anyx∈C, thesize|x|ofxis its indexind(x).

The map| − |:C→Nis thesize functionofC.

We say thatCis

connectedif#C(0) = 1;

augmentedifC(0) =∅;

monatomicifCis augmented and#C(1) = 1;

the unit collectionifCis connected andC=C(0);

the neutral collectionifCis monatomic andC=C(1).

Generating series

WhenCis a combinatorial graded collection, thegenerating seriesofCis the formal power series

GC(t) :=X

n>0

#C(n)tⁿ.

This series satisfies

G_C(t) =X

x∈C

t^|x|.

The generating series ofCencodes theinteger sequenceassociated withC, which is the sequence

(#C(0),#C(1),#C(2), . . .).

Multigraded collections

Letk>1. Ak-multigraded collectionis anN^k-collection.

Astatisticson anI-collectionCis a maps :C→N.

Given ak-multigraded collectionC, we obtain for anyj ∈[k]the statisticss_jdefined, for any x∈Cby

sj(x) :=nj

ifind(x) = (n₁, . . . , n_k).

WhenCis combinatorial, thegenerating seriesofCis the multivariate formal power series GC(t1, . . . , tk) := X

(n1,...,nk)∈N^k

#C((n1, . . . , nk)) tⁿ₁¹. . . tⁿ_k^k.

Colored collections

LetCbe a finite set, calledset of colors.

AC-colored collectionis anI_C-collection where

IC:={(a, u) :a∈Candu∈C^∗}, whereC^∗is the set of all finite words onC.

LetCbe aC-colored collection,x∈C, and assume thatind(x) = (a, u). Then theoutput colorout(x)ofxisa;

thei-th input colorin_i(x)ofxis thei-th letteru_iofufor anyi∈[|u|];

thesize|x|ofxis the length as a word ofin(u).

In particular, anyC-colored collectionCgives rise to a graded collectionC⁰whereC⁰(n) contains all elements ofCof sizen>0.

Section

2.2 Operations on collections

Casting

LetCbe anI-collection,J be an index set, and ω:I→J

be a map.

Theω-castingofCis theJ-collectionCast^ω(C)satisfying (Cast^ω(C)) (j) = [

i∈I ω(i)=j

C(i)

for anyj∈J.

WhenJ =N, one can seeω:I→Nas a size function wherein the size of anyx∈C(i)isω(i), and we say thatCast^ω(C)is theω-graduationofC.

Disjoint union

LetC1andC2be twoI-collections.

Thedisjoint unionofC₁andC₂is theI-collectionC₁tC₂satisfying (C1tC2) (i) =C1(i)tC2(i)

for anyi∈I.

– Proposition –

IfC1andC2are two combinatorialI-collections, thenC1tC2is combinatorial.

If, additionally,I=N, then

GC₁tC₂(t) =GC₁(t) +GC₂(t).

Cartesian product

LetC1be anI1-collection, . . . , andCpbe anIp-collection wherep>0.

TheCartesian productofC₁, . . . ,C_pis theI₁× · · · ×I_p-collection[[[C₁, . . . , C_p]]]_×satisfying [[[C1, . . . , Cp]]]_×((i1, . . . , ip)) =C1(i1)× · · · ×Cp(ip)

for any(i1, . . . , ip)∈I1× · · · ×Ip.

Whenω:I₁× · · · ×I_p→Jis a map, theω-Cartesian productofC₁, . . . ,C_pis theJ-collection [[[C1, . . . , Cp]]]^ω_×:=Cast^ω([[[C1, . . . , Cp]]]_×).

Cartesian product

When allC₁, . . . ,C_pare graded collections, we denote by+ :N^p→Nthe map satisfying + (n1, . . . , np) :=n1+· · ·+np.

In this case,[[[C1, . . . , Cp]]]⁺_×is a graded collection.

– Proposition –

IfC1be a combinatorialI1-collection, . . . , andCpbe a combinatorialIp-collection, then[[[C1, . . . , Cp]]]×is combinatorial.

If additionally,I1=· · ·=Ip=N, then

G[[[^C1,...,C_p]]]⁺_×(t) = Y

j∈[p]

GC_j(t).

List

LetCbe anI-collection.

Thelist collectionofCis theI^∗-collection List(C) := G

p>0

[[[C, . . . , C]]]×

| {z } pterms

.

Whenω:I^∗→Jis a map, theω-list collectionofCis theJ-collection List^ω(C) :=Cast^ω(List(C)).

– Proposition –

IfCis a combinatorial thenList(C)is combinatorial.

If additionally,I=NandCis augmented, then

G_List+(C)(t) = 1 1− GC(t).

Composition

LetC₁andC₂be two graded collections.

ThecompositionofC₁andC₂is the graded collectionC₁C₂satisfying

(C1C2) (n) = G

p>0

















C1(p),[[[C2, . . . , C2]]]⁺_×

| {z } pterms

(n)



















π₂

×

for anyn∈N, whereπ₂:N²→Nis the map defined byπ₂(n₁, n₂) :=n₂.

– Proposition –

IfC1andC2are two combinatorial and graded collections, andC2is augmented, thenC1C2is combinatorial.

Moreover,

GC₁C₂(t) =GC₁(GC₂(t)).

Section

2.3 Some collections

Words

Analphabetis a setAwherein elements are calledletters.

We can seeAas a graded collection where all letters have size1. ThenA=A(1).

The collections ofwordsonAis the graded collection A^∗:=List⁺(A).

We denote byu1. . . uneach element(u1, . . . , un)ofList⁺(A)(n). The size of a word is its length.

– Example –

LetA:={a,b,c}.

One hasA^∗(0) ={},A^∗(1) ={a,b,c},A^∗(2) ={aa,ab,ac,ba,bb,bc,ca,cb,cc}.

SinceG_A(t) = 3t,

G_A∗(t) = 1

1−3t=X

n>0

3ⁿtⁿ.

Integer compositions

We can seeNas a graded collection whereinN(n) ={n}for anyn>0. The collection ofinteger compositionsis the graded collection

Com :=List⁺(N\ {0}).

An integer composition is a sequenceλ=λ1. . .λkof positive integers. The size ofλis λ₁+· · ·+λ_k. Thelengthofλis its numberkofparts.

SinceG_N\{0}(t) =_1−t^t ,

GCom(t) = 1

1−_1−t^t = 1 +X

n>1

2ⁿ⁻¹tⁿ.

Theribbon diagramofλis the diagram obtained by concatenating lines of boxes for each part of λ.

– Example –

The integer compositionλ:= 3141admits the ribbon diagram

.

Binary trees

The collection ofbinary treesis the graded collectionBT defined as the one satisfying the relation

BT ={ } thhh

{ },[BT ,BT ]⁺_×iii

+

×,

where is an object of size0calledleafand is an object of size1calledinternal node.

By definition, a binary treetis either the leaf ;

or an ordered pair( ,(t1,t2)), wheret1andt2are both binary trees.

– Example –

The pair

( ,(( ,( , )),( ,(,( ,(, )))))) is a binary tree.

Binary trees — size and representation

By definition ofBT , the size of a binary treetsatisfies

|t|=

(0 ift= ,

1 +|t1|+|t2| otherwise, wheret= ( ,(t1,t2)). Binary trees are drawn by putting the root on the top.

– Example –

The binary tree

( ,(( ,( , )),( ,(,( ,(, )))))) is drawn as

.

Binary trees – enumeration

Moreover, the generating series ofBT satisfies the quadratic algebraic equation GBT (t) = 1 +tGBT (t)².

The unique solution of this equation having a combinatorial meaning is GBT (t) = 1−√

1−4t

2t =X

n>0

cat(n)tⁿ.

Letω:N→N\ {0}be the map defined byω(n) :=n+ 1for anyn>0.

By setting

BT :=Cast^ω(BT ), we have, for anyn>1.

BT (n) = BT (n−1).

The size of a binary tree ofBT is its number of leaves.

Section

2.4 Exercises

About enumeration

– Exercise –

Show the stated formulas for the generating series for the operations of disjoint union, Cartesian product, list, and composition.

– Exercise –

LetCbe a combinatorial, graded, and augmented collection. LetMSet(C)be the graded collection of multisets onC, that are multisets*x1, . . . , xp+where p>0andxj∈Cfor allj∈[p]. The size of a multiset is the sum of the sizes of its elements w.r.t. the size function

ofC.

Prove that

G_MSet(C)(t) = Y

n>1

1 1−tⁿ

#C(n)

.

– Exercise –

LetCbe a combinatorial, graded, and augmented collection. LetSet(C)be the graded collection of sets on C, defined as the subcollection ofMSet(C)restrained to multisets without repeated elements.

Prove that

G_Set(C)(t) = Y

n>1

(1 +tⁿ)^#C(n).

Collection morphisms

– Exercise –

A Dyck path of sizen>0is a path inN²starting from(0,0)and ending at(2n,0), made of steps(+1,+1)and (+1,−1). By settingDas the graded collection of all Dyck paths, show that the collectionsBT andDare isomorphic.

– Exercise –

LetPerbe the graded collections of all permutations, that are bijectionsσ: [n]→[n], where the size of a permutation is the cardinality of its domain.

Letbst : Per→BT be the collection morphism wherein, for anyσ∈Per,bst(σ)is the binary tree obtained by inserting the letters ofσfrom the right to the left following the binary search tree insertion algorithm.

Prove that two permutationsσandσ⁰belong to the same fiber ofbstiff one can transformσintoσ⁰through the (nonoriented) moves

uacvbw←→ucavbw

whereu,v, andware any words of integers, anda,b, andcare letters satisfyinga<b<c.

This binary relation is thesylvester relation [Hivert, Novelli, Thibon, 2005].

Chapter

3. Treelike structures

Section

3.1 Syntax trees

Planar rooted trees

The collection ofplanar rooted treesis the graded collectionPRTdefined as the one satisfying the relation

PRT =

{ },List⁺(PRT)+

×, where is an object of size1calledinternal node.

By definition, a planar rooted tree is an ordered pair( ,(t1, . . . ,tk))wherek>0and allti, i∈[k], are planar rooted trees.

The size oft= ( ,(t₁, . . . ,t_k))satisfies

|t|= 1 +X

i∈[k]

|ti|.

– Example –

The planar rooted tree( ,(( , ),( , ),( ,(( , ))),( , )))is drawn as

Planar rooted trees and binary trees

The generating series ofPRTsatisfies

GPRT(t) =t+GPRT(t)², so thatGPRT(t) =GBT (t).

TheKnuth rotation correspondenceis the collection isomorphismφ: PRT→BT defined for any planar rooted treet:= ( ,(t1, . . . ,tk))by

φ(t) :=

( ift= ( , ), ( ,(φ(t1), φ(( ,(t2, . . . ,tk))))) otherwise.

– Example –

7−φ→

Right partial action

LetA:=N\ {0}and

·: PRT×A^∗→PRT

be the right partial monoid action defined for any planar rooted treet:= ( ,(t1, . . . ,tk))by t·u:=

(t ifu=, tu₁· u2. . . u_|u|

otherwise.

Ifu1∈/ [k], thent·uis not defined. Therefore,·is a partial action.

– Example –

Let

t:= ,

We have

t·1 = ,

t·21 = ,

t·23 = .

t·231 = ,

t·3 = .

Language of a tree

Lett∈PRT. Let

N(t) :={u∈A^∗:t·uis well-defined}

be thelanguageoft.

Some definitions:

Any word ofN(t)is anode;

A maximal element ofN(t)for the prefix order relation is aleaf;

A node oftwhich is not a leaf isinternal;

Therootoftis the node;

A nodeuis anancestorof a nodevifuis a prefix ofv;

A nodevis achildofuifv=uifor ani∈N;

Thedepth-firstorder is the lexicographic total order onN(t);

Thedegreedeg(t)oftis its number of internal nodes;

Thearityari(t)oftis its number of leaves;

Theheightht(t)oftismax{|u|:u∈ N(t)}.

Language of a tree

– Example –

Let

t:= .

Then,

N(t) ={,1,2,21,211,2111,2112,22,23,231,232,3}.

The leaves oftare1,2111,2112,22,231,232, and3.

The internal nodes oftare,2,21,211, and23.

The depth-first order4sorts the nodes oftas

4142421421142111421124224234231423243.

The degree oftis5, its arity is7, and its height is4.

Syntax trees

LetGbe an augmented graded collection.

AG-syntax treeis a planar rooted treettogether with a map λ_t:N (t)→G,

whereN (t)is the set of internal nodes oft, and such that for anyu∈ N (t), ifuhask>1 children, thenλ_t(u)∈G(k).

– Example –

LetG:={a,b,c,d}such that|a|= 1,|b|= 2,|c|= 2, and|d|= 3.

Let

t :=

be theG-syntax tree such thatλt() =d,λt(2) =d, λt(21) =a,λt(211) =b, andλt(23) =b.

This syntax tree is depicted as

d

a b

d

b .

Graded collections of syntax trees

Given an augmented graded collectionG, letS(G)be the graded collection of allG-syntax trees, where for anyt∈S(G),|t|:= ari(t).

Alternatively,S(G)satisfies the relation

S(G) ={} t[[[{ },GS(G)]]]^π_ײ, where is an object of size1and is an object of size0.

Therefore, the generating series ofS(G)satisfies

G_S(G)(t) =t+G_G G_S(G)(t) .

– Example –

LetG:={a,b,c,d}such that|a|= 1,|b|= 2,|c|= 2, and|d|= 3.

Since

G_G(t) =t+ 2t²+t³, the generating series ofS(G)satisfies

G_S(G)(t) =t+G_S(G)(t) + 2G_S(G)(t)²+G_S(G)(t)³.

Section

3.2 Operations on trees

Partial composition

For anyt∈S(G)(n)and anyi∈[n], thei-th leafoftis thei-th leaf w.r.t. the depth-first order of t.

Let forn, m>1andi∈[n]the product

◦^(n,m)_i : S(G)(n)×S(G)(m)→S(G)(n+m−1)

such that, for anyt∈S(G)(n)ands∈S(G)(m),t◦^(n,m)_i sis theG-syntax tree obtained by overlying the root ofsonto thei-th leaf oft.

– Example –

c

a b

c b

1 5

8

◦^(8,5)₅ a

b

c =

c

b c

b a b

c

a

We shall omit the mention tonandmin◦^(n,m)_i in order to write simply◦i.

Full composition

Let for anyn>1andm₁, . . . , m_n>0the product

◦^(m1,...,mn): S(G)(n)×S(G) (m₁)× · · · ×S(G) (m_n)→S(G) (m₁+· · ·+m_n)

defined, for where for anyt∈S(G)(n),s₁∈S(G) (m₁), . . . ,s_n∈S(G) (m_n), by

◦^(m1,...,mn)(t,s₁, . . . ,s_n) := (. . .((t◦_ns_n)◦_n−1s_n−1). . .)◦₁s₁.

– Example –

◦^(2,1,4,2)









 ^a c

, ^a , , c

a , ^b











= a

a b c c a

We shall omit the mention tom₁, . . . ,m_nin◦^(m1,...,mn)in order to write simply◦.

Factors, prefixes, and suffixes

Lett,s∈S(G).

We say thatsis afactoroftif there existr,r₁, . . . ,r_|s|∈S(G)andi∈[|r|]such that t=r◦i s◦

r1, . . . ,r_|s|

. This property is denoted bys4ft.

Ifr=, thensis aprefixoft. This property is denoted bys4pt.

If allr_i=, thensis asuffixoft. This property is denoted bys4st.

– Example –

c

b 4f a b

a b

c c

b

b , ^c

b b

4p a

b

a b

c c

b

b ,

b b 4s

a b

a b

c c

b b

Occurrences

A nodeuoftis anoccurrenceofsintifsis a prefix oft·u. This nodeuis the position of the root ofsint.

– Example –

Given the tree

a

a a

a a b

a c b

,

the nodes1,3, and31are occurrences of

a a

and the nodes2and112are occurrences of

b .

By definition, a treesadmits an occurrence intiffs4ft.

Ifsadmits no occurrence int, thentavoidss.

Section

3.3 Term rewrite systems

Rewrite relations

Arewrite relationonS(G)is a binary relation→onS(G)such that ift→t⁰, then|t|=|t⁰|. Each pair(t,t⁰)such thatt→t⁰is arewrite rule.

Theclosureof→is the binary relation⇒onS(G)satisfying r◦i s◦

r1, . . . ,r_|s|

⇒r◦i s⁰◦

r1, . . . ,r_|s|

ifs→s⁰, whererandr₁, . . . ,r_|s|are anyG-trees, andi∈[|r|].

– Example –

If→is the rewrite relation satisfying b a

a

→

a b

b

,one has

a

a

a a

a a a

a

a

b ⇒ ^{a b}

a a

b a a

a

a a

.

Rewrite relations — definitions

Let→be a rewrite relation onG-trees (and⇒be its closure).

Let us define

⇒∗ as the reflexive and transitive closure of⇒;

⇔∗ as the reflexive, symmetric, and transitive closure of⇒.

A treetrewritesinto a treet⁰ift⇒^∗ t⁰. Two treestandt⁰arelinkedift⇔^∗ t⁰.

LetE_→be the graded collection of all⇔-equivalence classes.^∗ Anormal formfor→is a treetsuch thatt⇒^∗ t⁰impliest=t⁰. LetF→be the graded collection of all normal forms for→.

Termination and confluence

When there is no infinite chaint0⇒t1⇒t2⇒ · · ·,the rewrite relation→isterminating.

Whent⇒^∗ s₁andt⇒^∗ s₂implies the existence oft⁰such thats₁⇒^∗ t⁰ands₂⇒^∗ t⁰,→is confluent.

Whent⇒s1andt⇒s2implies the existence oft⁰such thats1

⇒∗ t⁰ands2

⇒∗ t⁰,→islocally confluent.

– Theorem (Diamond property)

–

If→is terminating and locally confluent, then→is confluent.

– Proposition –

If→is terminating, thenF→is the set of allG-trees avoiding the left members of→.

If→is terminating and confluent, thenF→is isomorphic, as a graded collection, withE→.

Some rewritings

– Example –

LetG:={a,b,c}with|a|=|b|= 2and|c|= 3, and→be the rewrite relation satisfying

a → b , _a ^b → c .

One has

a a

c a

b

a a

c b

b

c

c b

a

c

c b

b

a b

c a

b

a c

c

b b

c c b

This rewrite relation→isnot confluent.

It isterminating. This is implied by the fact that each rewriting decreases by one the number of internal nodes labeled byaand there is a finite number ofG-trees with a given arity.

An important rewrite relation: Tamari lattices

LetG:={a}with|a|= 2, and→be the rewrite relation onS(G)satisfying

a

a → ^a

a .

First graphs(S(G)(n),⇒):

a a

a

a a

a a

a

a a

a

a a

a a

a a

a a

a

a a

a a

a a

a a

a a

a a

a a

a a

a a

a a

a a

a a

a a

a a

a a

a a

a a

a a a

a a

a

a a

a a a

a a

a

a a

a a a

a a

a

1 2 3 4 5

Tamari rewrite relation

The binary relation→is theright rotation operation, an important operation appearing in computer science.

This rewrite relation is terminating and confluent.

As consequence, the binary relation⇒^∗ endow eachS(G)(n)with a poset structure, called Tamari order, having a least and greatest element.

The setF_→contains allright comb trees, that are the trees avoiding

a a .

The integer sequence associated withE_→is1,1,1,1, . . . .

Properties of Tamari lattices

Introduced by Tamari in order to study nonassociative operations[Tamari, 1962].

The Tamari order relation⇒^∗ endows each setBT(n)with the structure of a lattice[Huang, Tamari, 1972].

Its number of intervals (thare are, pairs(t,t⁰)such thatt⇒^∗ t⁰) have been enumerated in[Chapoton, 2006].

These intervals can be encoded by interval-posets[Châtel, Pons, 2013].

The sets of all Tamari intervals forms also lattice, the lattice of cubic coordinates, having a lot of combinatorial and geometrical properties[Combe, 2019].

Some generalizations have been introduced:

m-Tamari lattices[Bergeron, Préville-Ratelle, 2012]; ν-Tamari lattices[Préville-Ratelle, Viennot, 2017]; δ-canyon lattices[Combe, G., 2020].

A variant of Tamari lattices

LetG:={a}with|a|= 2, and→be the rewrite relation onS(G)satisfying

a a

a

→

a a

a

.

First graphs(S(G)(n),⇒):

a a

a

a a

a a

a a

a a

a a

a a

a a

a a

a

a a

a a

a a

a a

a a

a a

a a

a a a

a a

a

a a

a a

a a

a

a a

a a

a a

a a

a

a

a a

a

a a

a a a

a a

a a

a a

a

a a

a a

1 2 3 4 5

Properties of a variant of Tamari lattices

This rewrite relation is terminating butnot confluent.

TheBuchberger completion algorithm[Dotsenko, Khoroshkin, 2010]is a semi-algorithm taking as input a terminating an not confluent rewrite relation→, and outputting a new rewrite relation

→⁰such that

→⁰is terminatingand confluent;

the reflexive and transitive closures of⇒and⇒⁰are equal.

– Theorem

–

The collectionF→can be described as the set of theG-trees avoiding11trees.

The integer sequence associated withE→is

1,1,2,4,8,14,20,19,16,14,14,15,16,17, . . .

and its generating series satisfies GE→(t) = t

(1−t)² 1−t+t²+t³+ 2t⁴+ 2t⁵−7t⁷−2t⁸+t⁹+ 2t¹⁰+t¹¹ .