gebras
2
Contents
3
1. Introduction 1
4
2. Binary algebras 3
5
2.1. Polynomials 4
6
2.2. Sub-objects 4
7
2.3. Congruence preserving functions 5
8
3. Length condition 5
9
4. The toolbox 7
10
4.1. Congruent substitutions 7
11
4.2. Canonical representatives 8
12
4.3. Strong irreducibility 9
13
5. Proof of the main Theorem 10
14
5.1. The induction hypothesis 10
15
5.2. Partial polynomiality of CP functions 10
16
5.3. Polynomiality of CP functions 11
17
6. The case of trees 12
18
6.1. Canonical representative 13
19
6.2. Strongly irreducible trees 13
20
7. The case of words 13
21
7.1. Canonical representative 14
22
7.2. Strongly irreducible words 14
23
7.3. Application to free commutative monoids 16
24
8. Conclusion 16
25
References 17
26
1. Introduction
27
A function on an algebra is congruence preserving if, for any congruence, it
28
maps pairs of congruent elements onto pairs of congruent elements.
29
A polynomial function on an algebra is any function defined by a term
30
of the algebra using variables, constants and the operations of the algebra.
31
Obviously, every polynomial function is congruence preserving. An algebra
32
is said to be affine complete if every congruence preserving function is a
33
polynomial function.
34
We proved in [3] that if Σ has at least three elements, then the free
35
monoid Σ∗ generated by Σ is affine complete. If Σ has just one lettera, then
36
the free monoid a∗ is isomorphic to hN,+i, and we proved in [2] that, e.g.,
37
f: N → N defined by f(x) = if x == 0 then 1 else bex!c, where e =
38
2.718. . . is the Euler number, is congruence preserving but not polynomial.
39
Thus hN,+i, or equivalently the free monoid a∗ with concatenation, is not
40
affine complete. Intuitively, this stems from the fact that the more generators
41
Σ∗has, the more congruences it has too: thusNwith just one generator, has
42
very few congruences, hence many functions, including non polynomial ones,
43
can preserve all congruences ofN. We also proved in [1] that, when Σ has three
44
letters, in the algebra of full binary trees with leaves labelled by letters in
45
Σ, every unary CP function is polynomial. These previous works left several
46
open questions. What happens if Σ has one or two letters: for algebras of
47
trees? for non unary CP functions on trees? for the free monoid generated
48
by two letters? We answer these three questions in the present paper: these
49
algebras are affine complete.
50
For full binary trees and at least three letters in Σ, the proof of [1]
51
consisted in showing that CP functions which coincide on Σ are equal, and in
52
building for any CP functionf a polynomialPf such thatf(a) =Pf(a) for
53
a∈Σ, wherefrom we inferred that f =Pf for anyt. We now generalize this
54
result in three ways: we consider arbitrary trees (with labelled leaves) where
55
the empty tree is allowed, the alphabet Σ may have one or two letters instead
56
of at least three, and CP functions of any arity are allowed. Our method
57
mostly uses congruences∼u,v which substitute for occurrences of a treeua
58
smaller treev: in fact, we even restrict ourselves to congruences such that u
59
belongs to a subsetT which is chosen in a way ensuring that every congruence
60
class has a unique smallest canonical representative. Using these congruences,
61
we build, for each CP functionf, andτ∈ T, a polynomial Pτ such that, for
62
trees u1, . . . , un small enough, f(u1, . . . , un) = Pτ(u1, . . . , un). We finally
63
show that polynomials which coincide on Σ coincide on the whole algebra,
64
wherefrom we conclude that all thePτ are equal andf is a polynomial.
65
The next question is: is{a, b}∗equipped with concatenation affine com-
66
plete? We show in the present paper that the answer is positive. The essential
67
tool used in [3] was the notion of Restricted Congruence Preserving functions
68
(RCP), i.e., functions preserving only the congruences defined by kernels of
69
endomorphismshΣ∗,·i → hΣ∗,·i, which allowed to prove that RCP functions
70
are polynomial, implying that a fortiori CP functions are polynomial. Unfor-
71
tunately, the fundamental propertyP below, which was implicitly used when
72
there are three letters, no longer holds where there are only two letters.
73
(P)
Letγa,b be the homomorphism substituting bfora, if f: Σ→Σ is such that for alla, b ∈Σ,γa,b(f(a)) =γa,b(f(b)) then
f is either a constant function, or the identity.
74
Let Σ ={σ1, . . . , σn}. Whenn= 2, alas, property (P) is no longer true
75
and restricting ourselves to RCP functions cannot help in proving that CP
76
functions are polynomial. For instance, the functionf: Σ∗→Σ∗ defined by
77
f(w) = σ|w|1 σ1· · ·σ|w|n σn, where |w|σ denotes the number of occurrences of
78
the letterσin w, is clearly neither polynomial, nor CP (the congruence “to
79
have the same first letter” is not preserved). Fortunatelyf is not RCP when
80
n≥3, and thus is not a counter-example to the result stated in [3], but it is
81
RCP whenn= 2. Thus, for words in Σ∗, we here have to use a new method,
82
which also works even when|Σ|= 2 and which is very similar to the method
83
used for trees, even though the proofs are more complex to take into account
84
the associativity of the product (usually called concatenation) of words.
85
Most of the proofs of intermediate Lemmas and Propositions are iden-
86
tical for trees and for words or have only minor differences. Important differ-
87
ences, related to the associativity or non associativity of the product in the
88
corresponding algebras, are located in the the proofs of just two Assumptions,
89
that we prove separately.
90
The paper is thus organized as follows. In section 2, we recall the basics
91
about algebras, polynomials and congruence preserving functions. In Section
92
3 we prove that the relation between the length of the value of a function and
93
the length of its arguments is affine for both CP functions and polynomials.
94
In Section 4 we define the main kind of congruences we will use and we show
95
how to compute canonical representatives for these congruences. In section
96
5, we define polynomials associated with a CP function and prove that CP
97
functions are polynomial under two Assumptions given in the previous sec-
98
tion. In Section 6 (resp. 7) we prove these two Assumptions for the algebra of
99
trees (resp. the free monoid). Section 7 ends with an application of the result
100
on lengths of Section 3 which immediately implies the affine completeness of
101
the free commutative monoid.
102
2. Binary algebras
103
Let Σ be a nonempty finite alphabet, whose letters will be denoted by
104
a, b, c, d, . . ..
105
We consider an algebraic structurehA(Σ), ?,0i, with0∈/ Σ, subsuming
106
both the free monoid and the set of binary trees, satisfying the following
107
axioms (Ax-1), (Ax-2), (Ax-3)
108
(Ax-1) Σ∪ {0} ⊆ A(Σ),
109
(Ax-2) if u /∈Σ∪ {0}then∃v, w∈ A(Σ) :u=v ? w.
110
(Ax-3) there exists a mapping| · |:A(Σ)→Nsuch that
111
– |0|= 0,
112
– |σ|= 1, for allσ∈Σ,
113
– |u ? v|=|u|+|v|.
114
|u| is said to be the length ofu(it is equal to the number of occurrences of
115
letters of Σ inu). We similarly define, forσ∈Σ andu∈ A(Σ),|u|σ which is
116
the number occurrences of the letterσinu.
117
The free monoid and the algebra of binary trees are examples of such an
118
algebra. IfA(Σ) is the set of words Σ∗on the alphabet Σ,?is the (associative)
119
concatenation of words, and0is the empty wordε, we get the free monoid. If
120
A(Σ) is the set of binary trees whose leaves are labelled by letters of Σ,t ? t0
121
is a tree consisting of a root whose left subtree ist and whose right subtree
122
ist0, and0is the empty tree then we get the algebra of binary trees. In the
123
case of trees the operation?is not associative. The free commutative monoid
124
hNp,+,(0, . . . ,0)iis also a binary algebra satisfying (Ax-1), (Ax-2), (Ax-3).
125
For our proofs the main difference between trees and the other examples
126
relates to point (Ax-2) above: the decompositionu=v ? wis unique for trees
127
and not for the other examples.
128
Fact 2.1 (Unicity of decomposition). If t is a tree not in{0} ∪ Σthen there
129
exists a unique ordered pairht1, t2i 6=h0,0iinA2 such that t=t1? t2.
130
An element ofA(a word or a tree) will be called an object.
131
2.1. Polynomials
132
We denote by A the set A(Σ). We also consider the infinite set of vari-
133
ables X = {xi |i ≥ 1}, disjoint from Σ. We denote by An, the set A(Σ∪
134
{x1, . . . , xn}). Note thatA=A0 and thatAn⊆ An+1.
135
Definition 2.2. An-ary polynomial with variables{x1, . . . , xn}is an element
136
P ofAn. The multidegree ofP is then-tuplehk1, . . . , kniwhereki=|P|xi.
137
With every such polynomial P we associate a n-ary polynomial function
138
P˜:An→ Adefined by:
139
for any~u=hu1, . . . , ui, . . . , uni ∈ An,
140
P(~˜ u) =
P ifP =0or P∈Σ
ui ifP =xi
Pf1(~u)?Pf2(~u) ifP =P1? P2
141
Note. In the case of words we have to prove that the value ofPeis independent
142
of its decompositionP =P1? P2. This is due to the fact thatPe(~u) can be
143
seen as a homomorphic image ofP by an homomorphism fromAn toA.
144
From now on we simply writeP instead of ˜P for denoting the function
145
associated with the polynomialP.
146
2.2. Sub-objects
147
Let A1,1 be the set of degree 1 unary polynomials with variable y, i.e., el-
148
ements P ∈ A(Σ∪ {y}) such that |P|y = 1, or objects of A(Σ∪ {y}) with
149
exactly one occurrence ofy.
150
Definition 2.3. An element u of A is a sub-object of an element t ∈ A, if
151
there exists an occurrence ofuinsidet, formally: if there exists a polynomial
152
P ∈ A1,1 such thatP(u) =t.
153
In the case of words (resp. trees), sub-objects are factors (resp. subtrees).
154
Definition 2.4. A sub-polynomialQ of a polynomialP ∈ An is a sub-object
155
ofP.
156
2.3. Congruence preserving functions
157
Definition 2.5. A congruence onhA, ?,0iis an equivalence relation∼ com-
158
patible with?, i.e.,s1∼s01 ands2∼s02 implys1? s2∼s01? s02.
159
Definition 2.6. A function f: An → A is congruence preserving (abbre-
160
viated into CP) on hA, ?,0i if, for all congruences ∼ on hA, ?,0i, for all
161
t1, . . . , tn, t01, . . . , t0n inA, ti ∼t0i for alli= 1, . . . , n, implies f(t1, . . . , tn)∼
162
f(t01, . . . , t0n).
163
Obviously, every polynomial function is CP. Our goal is to prove the
164
converse, namely
165
Theorem 2.7. Assume|Σ| ≥2for words and|Σ| ≥1for trees. Iff:A(Σ)n→
166
A(Σ)is CP then there exists a polynomial Pf such that f =Pff.
167
This is the main result of the paper, which will be proven in Sections 5, 6
168
and 7.
169
3. Length condition
170
For polynomials, as a consequence of (Ax-3), we get:
171
Fact 3.1. If P ∈ An is a polynomial of multidegreehk1, . . . , knithen
172
|P(u1, . . . , un)|=|P(0, . . . ,0)|+Pn
i=1ki.|ui|.
173
A necessary condition for a function f: An → A to be polynomial is
174
that f has in someway a multidegree hk1, . . . , kni, playing the rˆole of the
175
multidegree of polynomials, i.e., such that |f(u1, . . . , un)| =|f(0, . . . ,0)|+
176
Pn
i=1ki.|ui|.For words when |Σ| ≥3, the existence of such a multidegree is
177
proved in [3]. We here generalise this proof so that it also applies to trees and
178
to smaller alphabets.
179
Lemma 3.2. Letf:A(Σ)n → A(Σ) be an-ary CP function.
180
(1) There exist functions λ, λi:Nn → N such that |f(u1, . . . , un)| =
181
λ(|u1|, . . . ,|un|)and|f(u1, . . . , un)|i=λi(|u1|i, . . . ,|un|i), for i= 1,2.
182
(2)λ(p1+q1, . . . , pn+qn) =λ1(p1, . . . , pn) +λ2(q1, . . . , qn).
183
Proof. For an objectu∈ A, denote by|u|1=|u|a the number of occurrences
184
of the letter a in u, and let |u|2 = |u| − |u|1. Formally, |ε|1 = 0, |a|1 = 1,
185
|σ|1= 0 forσ6=a, and|t ? t0|1=|t|1+|t0|1.
186
(1) As the relation |u| = |v| is a congruence and f is CP, |ui| = |vi|
187
fori= 1, . . . , nimplies|f(u1, . . . , un)|=|f(v1, . . . , vn)|hence|f(u1, . . . , un)|
188
depends only on the lengths|u1|, . . . ,|un|, andλis well defined. Similarly for
189
λi, i= 1,2 as|u|i =|v|i is also a congruence.
190
(2) Consider objectsui with|ui|1=piand|ui|2=qi(see Figure 1). On
191
the one hand, |f(u1, . . . , un)| = λ(|u1|, . . . ,|un|) = λ(p1+q1, . . . , pn+qn),
192
|f(u1, . . . , un)|1 = λ1(p1, . . . , pn) and |f(u1, . . . , un)|2 = λ2(q1, . . . , qn). On
193
the other hand, |f(u1, . . . , un)| = |f(u1, . . . , un)|1+|f(u1, . . . , un)|2, hence
194
(2).
195
HH
HH HH
HH H
HH HH
HH HH
H H HH a
c a
c
b a
c
Figure 1. A treeui withpi =|ui|1= 3 andqi=|ui|2= 4.
Proposition 3.3. For anyn-ary CP functionf:A(Σ)n → A(Σ), with|Σ| ≥2,
196
there exists a n-tuplehk1, . . . , kniof natural numbers, called the multidegree
197
off, such that |f(u1, . . . , un)|=|f(0, . . . ,0)|+Pn
i=1ki.|ui|.
198
Proof. Lete~i=h
(i−1) times
z }| {
0, . . . ,0 ,1,0, . . . ,0i,~0 =h0, . . . ,0i, and apply Lemma 3.2.
We have for anym1, . . . , mi, . . . , mn,
λ(m1, . . . , mi+ 1, . . . , mn) =λ1(m1, . . . , mi, . . . , mn) +λ2(~ei), λ(m1, . . . , mi, . . . , mn) =λ1(m1, . . . , mi, . . . , mn) +λ2(~0).
Subtracting
λ(m1, . . . , mi+ 1, . . . , mn)−λ(m1, . . . , mi, . . . , mn) =λ2(~ei)−λ2(~0).
Setting ki=λ2(~ei)−λ2(~0), we get
λ(m1, . . . , mi, . . . , mn)−λ(m1, . . . , mi−1, . . . , mn) =ki
... λ(m1, . . . ,1, . . . , mn)−λ(m1, . . . ,0, . . . , mn) =ki
Summing up λ(m1, . . . , mi, . . . , mn)−λ(m1, . . . ,0, . . . , mn) =kimi
Iterating for all i, λ(m1, . . . , mn)−λ(~0) =k1m1+· · ·+knmn. Proposition 3.3 holds both for words and trees. However, for trees the
199
following better result holds even when|Σ|= 1.
200
Proposition 3.4.In the algebra of trees, for anyn-ary CP functionf:A(Σ)n→
201
A(Σ), there exists an-tuplehk1, . . . , kniof natural numbers, called the multi-
202
degree of f, such that|f(u1, . . . , un)|=|f(0, . . . ,0)|+Pn
i=1ki.|ui|.
203
Proof. For a treeu /∈ Σ,|u|1 (resp.|u|2) is the number of left (resp. right)
204
leaves, so that|u|=|u|1+|u|2foru /∈Σ. On Figure 1|ui|1= 4 and|ui|2= 3.
205
Formally,|0|=|0|1=|0|2= 0. Foru=t ? t0 ∈/ Σ we have
206
|u|1=|t0|1+
1 ift∈Σ,
|t|1 ift /∈Σ. and|u|2=|t|2+
1 ift0 ∈Σ,
|t0|2 ift0 ∈/ Σ.
207
We already know that the relation∼defined byu∼v iff|u|=|v| is a
208
congruence. Forj= 1,2, the relation∼jdefined byu∼j viff eitheru=v∈Σ
209
oru, v /∈Σ and|u|j =|v|j is a congruence. Hence iff =An→ Ais CP then
210
for all u1, . . . , un, v1, . . . , vn ∈/ Σ such that ∀i = 1, . . . , n,|ui|j = |vi|j and
211
f(u1, . . . , un), f(v1, . . . , vn)∈/ Σ, we have |f(u1, . . . , un)|j =|f(v1, . . . , vn)|j.
212
Without loss of generality, we may assume that for allu1, . . . , un,f(u1, . . . , un)
213
is not in Σ. This holds because g(u1, . . . , un) = 0? f(u1, . . . , un) is CP and
214
|g(u1, . . . , un)|=|f(u1, . . . , un)|.
215
Foru /∈Σ,|u|=|u|1+|u|2. Exactly as in Proposition 3.3 we show that
216
for anym1, . . . , mi, . . . , mn,λ(m1, . . . , mn)−λ(~0) =k1m1+· · ·+knmn.It fol-
217
lows that for allu1, . . . , un ∈/ Σ,|f(u1, . . . , un)|=|f(0, . . . ,0)|+Pn
i=1ki.|ui|.
218
Finally, as for all u ∈ A, u ? 0 ∈/ Σ and |u ?0| = |u|, we have:
219
|f(u1, . . . , un)|=|f(u1?0, . . . , un?0)| =|f(0, . . . ,0)|+Pn
i=1ki.|ui?0| =
220
|f(0, . . . ,0)|+Pn
i=1ki.|ui|.
221
4. The toolbox
222
4.1. Congruent substitutions
223
If f is CP then f(u) ∼ f(v) as soon as u ∼ v. This is why we introduce
224
specific congruences∼u,v such thatu∼u,vv, so that if for some polynomial
225
Q, (which is also CP), we know that for someu,f(u) =Q(u), then we know
226
that for allv,f(v)∼u,vQ(v). Thus it is important to describe the congruence
227
classes of such congruences.
228
Definition 4.1. For u, v a couple of objects in A the relation ∼u,v is the
229
equivalence relation generated by the set of pairs{hP(u), P(v)i |P ∈ A1,1}.
230
∼u,v is clearly a congruence onhA, ?,0i.
231
Given such a congruence, we can consider the quotient algebra. It may
232
happen that each congruence class has a simple canonical representative.
233
For instance, the canonical representative could be the shortest object in
234
the congruence class, provided it is unique. However unicity of the shortest
235
representative certainly does not hold for the congruences∼u,vwhen|u|=|v|.
236
It also happens that unicity does not hold even when|u|>|v|(Remark 4.2).
237
Remark 4.2. Even if |u| > |v|, there might be several shortest congruent
238
elements. For instance in the case of words,ab∼aa,baaa∼aa,bba, henceab
239
andbaare two shortest elements congruent toaaa.
240
Definition 4.3. For a given elementτ ofA, an elementt∈ A isτ-reducible,
241
ifτ is a sub-object oft. We denote by Θτ the set of allτ-irreducible objects
242
inA.
243
In Figure 2,Qτ isτ-reducible,QandPτareτ-irreducible, and in Figure
244
3,t00isτ-irreducible.
245
We now extend Definition 4.3 of τ-irreducible objects in Ato polyno-
246
mials inAn.
247
τ =q
c A
A d
Q=q q
c
A A q
x A
A d
Qτ= q
b A
A x1
H H q
c A
A d
q Pτ = q
q
b A
A x1
A A
x2
Figure 2. From left to right: treeτ =c ? d, aτ-irreducible polynomialQwith variable x, aτ-reducible polynomialQτ
with variable x1 together with its associated τ-irreducible polynomialPτ =Red∗τ,x2(Qτ).
Definition 4.4. Letτ ∈ A. A polynomial P ∈ An is said to beτ-irreducible
248
if any sub-objectv ofP which is inAisτ-irreducible.
249
Intuitively, the constant sub-objects (“coefficients”) ofPareτ-irreducible.
250
In Figure 2,Qτ is the onlyτ-reducible polynomial.
251
4.2. Canonical representatives
252
In fact it is possible to define and to “compute” a canonical representative
253
t0 oft for∼τ,v if|τ|>|v|. To this end we stepwise replace every occurrence
254
of τ inside t by v. To make this process deterministic we define the reduct
255
Redτ,v(t) obtained by replacing byvthe “leftmost” occurrence ofτ inside a
256
τ-reducible objectt.
257
Definition 4.5. (Definition of Redτ,v(t).)
258
Case of treesIft=τthenRedτ,v(t) =v. Oherwise, sincet6=τisτ-reducible,
259
|t|>|τ| ≥1, hence, by (Ax-2),t=t1? t2, and at least oneti isτ-reducible.
260
Either t1 ∈ Ais τ-reducible, and then Redτ,v(t) = Redτ,v(t1)? t2, or t1 is
261
τ-irreducible, thent2 isτ-reducible andRedτ,v(t) =t1? Redτ,v(t2). Figure 3
262
illustrates this reduction process.
263
Case of words Since τ is a factor of t, there exists a shortest prefix t0 of t
264
such thatt=t0τ t00. ThenRedτ,v(t) =t0vt00.
265
τ= q
c A
A d
t= q q
q
c A
A d
A A q
c A
A d
t0 = q q
a
A A q
c A
A d
t00= q q
a
A A
a
Figure 3. From left to right,τ=c?d,t= ((c?d)?0)?(c?d), t0= (a?0)?(c?d)) =Redτ,a(t),t00=Redτ,a(t0) = (a?0)?a).
We iterate this partial reduction function to get a mappingRed∗τ,v:A → Θτ inductively defined by:
Red∗τ,v(t) =
t ift∈Θτ
Red∗τ,v(Redτ,v(t)) ift /∈Θτ.
Proposition 4.6. Red∗τ,v(u ? w) =Red∗τ,v(Red∗τ,v(u)? w).
266
Proof. By definition, Red∗τ,v(t) = Redkτ,v(t), where k is the least integer
267
such that Redkτ,v(t) is τ-irreducible. If Red∗τ,v(u ? w) = Redpτ,v(u ? w) and
268
Red∗τ,v(u) = Redqu,v(u), necessarily q ≤ p and we have by induction on
269
i = 0, . . . , q, Redpτ,v(u ? w) = Redp−iτ,v(Rediτ,v(u)? w) hence the result for
270
i=q.
271
AlthoughRed∗τ,v(t) is a canonical representative of the congruence class
272
oftmodulo∼τ,v, it is not necessarily the only object of the equivalence class
273
ofthaving minimal length, as shown in Remark 4.2.
274
To prevent such situations, we will first define for each algebra a suitably
275
chosen subsetT of the algebra ensuring that for eachτ ∈ T, there exists a
276
unique canonical representative of shortest length in the class of ∼τ,v for
277
eachv ∈ A such that |v|< |τ| (Proposition 4.8). This set T has to satisfy
278
the following assumption.
279
Assumption 4.7. ∀τ∈ T, v∈ A, P ∈ A1,1, Red∗τ,v(P(τ)) =Red∗τ,v(P(v)).
280
Proposition 6.3 (resp. 7.1) shows that this assumption holds for the set
281
T of trees defined by (6.1) in Section 6 (resp. the setT of words defined by
282
(7.1) in Section 7).
283
Provided the truth of this assumption, we get:
284
Proposition 4.8. (Existence of a canonical representative) Let τ ∈ T, and
285
v∈ Awith |τ|>|v|. For any t, t0∈ A,t∼τ,vt0 iff Red∗τ,v(t) =Red∗τ,v(t0).
286
Proof. By the definition ofRed∗τ,v, for allt, t0, t∼τ,v Red∗τ,v(t), andt0 ∼τ,v 287
Red∗τ,v(t0). HenceRed∗τ,v(t) =Red∗τ,v(t0) impliest∼τ,vt0 by transitivity.
288
Conversely, ift∼τ,vt0 then there existt1=t, t2, . . . , tn=t0, andPi∈
289
A1,1 (see Definition 4.1) such that for each i= 1, . . . , n−1,ti =Pi(τ) and
290
ti+1 =Pi(v) (or vice-versa). By Assumption 4.7,Red∗τ,v(ti) =Red∗τ,v(ti+1),
291
henceRed∗τ,v(t) =Red∗τ,v(t0).
292
Proposition 4.9. Let τ ∈ T, t and t0 be two objects such that |v| < |τ|,
293
t∼τ,vt0, and|t|<|τ|. Thent=t0 if and only if|t|=|t0|.
294
Proof. Ift =t0 then obviously |t| =|t0|. Sincet ∼τ,vt0, by Proposition 4.8,
295
Red∗τ,v(t) = Red∗τ,v(t0). But |t0| = |t| < |τ| implies that both t0 and t are
296
τ-irreducible, hencet=Red∗τ,v(t) =Red∗τ,v(t0) =t0.
297
4.3. Strong irreducibility
298
By Propositions 4.8 and 4.9, we get that if |t| < |τ| and |Red∗τ,v(t0)| > |τ|
299
thent6∼τ,ut0. To prove that if|t0|>|τ| then|Red∗τ,v(t0)|>|τ|, it is enough
300
to prove that if t0 contains a sub-object w of length n ≥ |τ| then w is a
301
sub-object ofRed∗τ,v(t0). This leads to the following definition.
302
Definition 4.10. Letτ ∈ A, an object w is said to be stronglyτ-irreducible
303
if |w| ≥ |τ| and if whenever w is a sub-object of some t ∈ A, w also is a
304
sub-object ofRed∗τ,v(t) for anyvsuch that|v|<|τ|.
305
We finally state the following assumption on T, the truth of which is
306
proven in Proposition 6.4 (resp. 7.3) for trees (resp. for words).
307
Assumption 4.11. For all τ ∈ T and for all τ-irreducible unary polynomials
308
P of degreeksuch that |τ| ≥2k+ 4, we have the following property:
309
If for allu∈ Asuch that |u| ≤1,P(u)isτ-reducible, then there exists
310
θ∈ A of length 1 and a strongly τ-irreducible sub-object wof P(θ)of length
311
not less than|τ|(i.e., |w| ≥ |τ|).
312
5. Proof of the main Theorem
313
From now on, we postulate the existence of a setT which satisfies Assump-
314
tions 4.7 and 4.11.
315
5.1. The induction hypothesis
316
The polynomiality of CP functions will be proved by induction on their arity.
317
The basic step of this induction is obvious and common to all algebras we
318
consider: a function of arity 0 is a constant, which is a polynomial function.
319
For the inductive step, note that if n ≥ 0 and f is a (n+ 1)-ary
320
CP function of multidegree hk1, . . . , kn, kn+1i, then for all t, ft defined by
321
ft(u1, . . . , un) = f(u1, . . . , un, t) is CP with multidegree hk1, . . . , kni, hence
322
the induction hypothesis:
323
Fact 5.1.
Induction hypothesis.For anyt∈ A, there exists a polynomial Qt of multidegree hk1, . . . , knisuch that:
∀u1, . . . , un ∈ A, Qt(u1, . . . , un) =f(u1, . . . , un, t).
324
Definition 5.2. ThepolynomialPτ associated withf andτ∈ T is the unique τ-irreducible polynomial of multidegreehk1, . . . , kn, misuch that
∀u1, . . . , un∈ A, Pτ(u1, . . . , un, τ) =Qτ(u1, . . . , un) =f(u1, . . . , un, τ).
It is also defined byPτ =Red∗τ,xn+1(Qτ), consideringPτ and Qτ as objects
325
inA(Σ∪ {x1, . . . , xn, xn+1}).
326
Figure 2 illustrates this definition in the algebra of binary trees.
327
5.2. Partial polynomiality of CP functions
328
Assuming the hypothesis stated in Fact 5.1, we can proceed and prove
329
Proposition 5.3. Letτ ∈ T. If|u|<|τ|and if |f(u1, . . . , un, u)|<|τ| then
330
• f(u1, . . . , un, u) =Red∗τ,u(Pτ(u1, . . . , un, u))
331
• eitherm=kn+1 andf(u1, . . . , un, u) =Pτ(u1, . . . , un, u), orm < kn+1
332
andPτ(u1, . . . , un, u)isτ-reducible.
333
Proof. Obviously, f(u1, . . . , un, u) ∼τ,u f(u1, . . . , un, τ) = Pτ(u1, . . . , un, τ)
334
∼τ,uPτ(u1, . . . , un, u). As|f(u1, . . . , un, u)|<|τ|,f(u1, . . . , un, u) isτ-irredu-
335
cible. Thus, by Assumption 4.7,f(u1, . . . , un, u) =Red∗τ,u(Pτ(u1, . . . , un, u)).
336
Let d = |f(u1, . . . , un, τ)| = |Pτ(u1, . . . , un, τ)|. Then |f(u1, . . . , un, u)| =
337
d−kn+1(|τ| − |u|) and|Pτ(u1, . . . , un, u)|=d−m(|τ| − |u|).
338
By Proposition 4.9,Pτ(u1, . . . , un, u) =f(u1, . . . , un, u) if and only if
339
|Pτ(u1, . . . , un, u)|=|f(u1, . . . , un, u)|if and only ifm=kn+1.
340
Sincef(u1, . . . , un, u) =Red∗τ,u(Pτ(u1, . . . , un, u)), if f(u1, . . . , un, u)6=
341
Pτ(u1, . . . , un, u) thenPτ(u1, . . . , un, u) is notτ-irreducible.
342
Hence d−m(|τ| − |u|) = |Pτ(u1, . . . , un, u)| ≥ |τ| > |f(u1, . . . , un, u)| =
343
d−kn+1(|τ| − |u|), which implies m < kn+1.
344
An immediate consequence of Proposition 5.3 is:
345
Proposition 5.4. Letτ∈ T, lethk1, . . . , kn, mibe the multidegree ofPτ. Then
346
(1) either m = kn+1 and for all u ∈ A such that |u| ≤ |τ|, and for all
347
u1, . . . , un∈ A such that|f(u1, . . . , un, u)|<|τ|, we have
348
Pτ(u1, . . . , un, u) =f(u1, . . . , un, u),
349
(2) or m < kn+1 and for all u ∈ A such that |u| ≤ |τ|, and for all
350
u1, . . . , un ∈ A such that |f(u1, . . . , un, u)| < |τ|, Pτ(u1, . . . , un, u) is
351
τ-reducible.
352
5.3. Polynomiality of CP functions
353
We first prove that for almost allτ we are in case (1) of Proposition 5.4.
354
Proposition 5.5. Let hk1, . . . , kn, kn+1ibe the multidegree off, let k=k1+
355
· · ·+kn+kn+1, and let τ ∈ T be such that τ ≥2k+ 4. For allu∈ A such
356
that|u|<|τ|and for allu1, . . . , un ∈ Asuch that|f(u1, . . . , un, u)|<|τ|, we
357
havePτ(u1, . . . , un, u) =f(u1, . . . , un, u).
358
Proof. By Proposition 5.4 it is enough to prove thatm < kn+1 is impossible.
359
LetPτ be theτ-irreducible polynomial associated withτ of multidegree
360
hk1, . . . , kn, miand let us assume thatm < kn+1. Then, by Proposition 5.4,
361
we have: for allu∈ Asuch that|u| ≤ |τ|and|f(u, . . . , u, u)|<|τ|, the object
362
Pτ(u, . . . , u, u) isτ-reducible.
363
We now consider theτ-irreducible unary polynomialPτ0 of degreeM =
364
k1+· · ·+kn+m < k, obtained by substitutingx1for any variablexi inPτ.
365
SincePτ0(u) isτ-reducible for allusuch that|u| ≤1<|τ|, by Assumption 4.11
366
there existθof length 1 and a stronglyτ-irreducible sub-objectwofPτ0(θ) =
367
Pτ(θ, . . . , θ, θ) of length not less thanτ. By Proposition 5.3,wis a sub-object
368
ofRed∗τ,θ(Pτ(θ, . . . , θ, θ)) =f(θ, . . . , θ, θ). Hence|w| ≤ |f(θ, . . . , θ, θ)|<|τ| ≤
369
|w|, a contradiction.
370
Letτ1andτ2be such that|τi|>|f(a, . . . , a)|. Then, by Proposition 5.5,
371
we have :
372
For allu1, u2, . . . , un, usuch that|u|and|f(u1, . . . , un)|are less that|τ1|and
373
|τ2|then
374
Pτ1(u1, . . . , un, u) = f(u1, . . . , un, u) =Pτ2(u1, . . . , un, u). (5.1) We first prove thatPτ1 =Pτ2 as a consequence of the next Proposition by
375
observing that equation (5.1) holds for allui,uof length 1.
376
Proposition 5.6. LetP,Qbe polynomials of multidegreehk1, . . . , kni.
377
If, for all u1, u2, . . . , un of length 1, P(u1, . . . , un) = Q(u1, . . . , un) then
378
P =Q.
379
Proof. For a polynomialP in the algebra of trees, we define s(P) to be the
380
number of symbols of Σ∪{?}∪{x1, . . . , xn}occurring inP. Formallys(0) = 0,
381
s(a) = 1 fora∈Σ∪ {x1, . . . , xn}, ands(u ? v) = 1 +s(u) +s(v). For P in
382
the algebra of words, we sets(P) =|P|.
383
In both cases there exists at least two distinct objects of length 1: either
384
two distinct lettersa, b, or the treesa ?0and0? a.
385
The proof is by induction ons(P).
386
Basis.
387
(1) Ifs(P) =s(Q) = 0 thenP =0=Q.
388
(2) If s(P) = s(Q) = 1 then P, Q ∈ Σ ∪ {x1, . . . , xn}. If P and Q are
389
both constants, the result follows from equalityP(u, . . . , u) =Q(u, . . . , u). If
390
P =xiandQ=xjwithi6=j, the hypothesisP(u1, . . . , un) =Q(u1, . . . , un)
391
leads to a contradiction, as soon asui 6=uj, hencei=j. IfP is a constant
392
uand Qis a variable xi, we have u=P(u0, . . . , u0) =Q(u0, . . . , u0) =u0, a
393
contradiction whenu6=u0.
394
Inductive step.If s(P)>1 then P =P1? P2 and Q=Q1? Q2, (tak-
395
ing |P1| = |Q1| = 1 in case of words). For any u1, u2, . . . , un of length 1,
396
we have Q(u1, . . . , un) =P(u1, . . . , un) =P1(u1, . . . , un)? P2(u1, . . . , un) =
397
Q1(u1, . . . , un)?Q2(u1, . . . , un) which impliesPi(u1, . . . , un) =Qi(u1, . . . , un),
398
hence, by the induction hypothesis,P1 =Q1 and P2 =Q2, and thus P =
399
Q.
400
Theorem 5.7. Letf be a CP function of multidegreehk1, . . . , kn, kn+1i. There
401
exists a polynomialPf of multidegreehk1, . . . , kn, kn+1isuch for allu1, . . . , un,
402
u∈ A,Pf(u1, . . . , un, u) =f(u1, . . . , un, u).
403
Proof. By Propositions 5.5 and 5.6 there exists a unique polynomial Pf
404
such that for allτ of length greater than |f(a, a, . . . , a)|, Pτ =Pf. For any
405
u1, . . . , un, u there exists τ such that |τ| > max(|u|,|f(u1, . . . , un, u)|). By
406
Proposition 5.5,f(u1, . . . , un, u) =Pτ(u1, . . . , un, u) =Pf(u1, . . . , un, u).
407
6. The case of trees
408
We here consider the algebra of binary trees with labelled leaves. For this
409
algebra of trees we set
410
T = {τ∈ A | |τ| ≥2} (6.1)
Proposition 6.1. If a tree wisτ-irreducible, then it is strongly τ-irreducible.
411
Proof. By definition ofRed∗τ,v, it is enough to show that ifw is a subtreee
412
of t then it is a subtree of Redτ,v(t). The proof is by induction on |t| such
413
thatwis a subtree oft. Iftisτ-irreducible thenRedτ,v(t) =tand the result
414
is proved. Otherwise,t=t1? t2, with wsubtree of someti, andRedτ,v(t) =
415
Redτ,v(t1)? t2or Redτ,v(t) =t1? Redτ,v(t2). In both cases,wis a subtree of
416
Redτ,v(t).
417
6.1. Canonical representative
418
For trees, we can improve Proposition 4.6.
419
Proposition 6.2. Red∗τ,v(u ? w) =Red∗τ,v(Red∗τ,v(u)? Red∗τ,v(w)).
420
Proof. By taking Proposition 4.6 into account, we just have to prove that
421
Red∗τ,v(u ? w) =Red∗τ,v(u ? Red∗τ,v(w)) whenuisτ-irreducible. This a conse-
422
quence of the definition of the leftmost reduction for trees:Redτ,v(u ? w) =
423
u ? Redτ,v(w).
424
We now prove that Assumption 4.7 holds for our algebra of binary trees.
425
Proposition 6.3. ∀P ∈ A1,1 Red∗τ,v(P(τ)) =Red∗τ,v(P(v)).
426
Proof. The proof is by induction on|P|. IfP =ythenRed∗τ,v(τ) =Red∗τ,v(v) =
427
v.
428
IfP =P1? P2 then by Proposition 6.2,
Red∗τ,v(P(τ)) =Red∗τ,v(Red∗τ,v(P1(τ))? Red∗τ,v(P2(τ))),and Red∗τ,v(P(v)) =Red∗τ,v(Red∗τ,v(P1(v))? Red∗τ,v(P2(v))).
Then, by the induction hypothesis, Red∗τ,v(Pi(v)) = Red∗τ,v(Pi(τ)), for i =
429
1,2, and thusRed∗τ,v(P(v)) = Red∗τ,v(P(τ)).
430
6.2. Strongly irreducible trees
431
The following Proposition assures that Assumption 4.7 holds for trees.
432
Proposition 6.4. For allτ ∈ T and for allτ-irreducible unary polynomials P
433
the following property holds.
434
If for allu∈ Asuch that |u| ≤1,P(u)isτ-reducible, then there exists
435
θ∈ Aof length 1 and a stronglyτ-irreducible subtreewofP(θ)of length not
436
less than|τ| (i.e., |w| ≥ |τ|).
437
Proof. Letτ ∈ T, which has length at least 2. Let P be a non constant τ-
438
irreducible polynomial such that for allu∈ A with length |u| ≤1, P(u) is
439
τ-reducible. Letσ∈Σ, and lett=σ ?0andt0 =0? σ,t6=t0.
440
AsP(t) is τ-reducible, it must contain τ. But sinceP is τ-irreducible,
441
there exists a non constant sub-polynomialQofP such thatQ(t) =τ. Then
442
|Q(t)| =|Q(t0)| =|τ| and, asQ is non-constant, Q(t0)6=τ. It follows that
443
Q(t0) isτ-irreducible, hence stronglyτ-irreducible by Proposition 6.1. We set
444
θ=t0 andw=Q(t0).
445
7. The case of words
446
For words, proving Assumptions 4.7 and 4.11 requires more work because
447
unicity of the decomposition fails in the free monoid.
448
As shown in Remark 4.2, Assumption 4.7 does not hold for any word τ.
449
Indeed, Assumption 4.7 fails as soon asτ self-overlaps, i.e., when there exists
450
a wordt which is a both a strict prefix and a strict suffix ofτ. For instance,
451
if τ = aba, ab ∼aba,ε ababa ∼aba,ε ba, while Redaba,ε(ab) = ab 6= ba =
452
Redaba,ε(ba). Obviously, words such that anbn do not self-overlap and thus
453
satisfy Assumption 4.7. But we also need that these words satisfy Assumption
454
4.11. The condition that τ is not self-overlapping is not sufficient to satisfy
455
Assumption 4.11. For instance, let τ = aabb and P = aax1bb, which is τ-
456
irreducible. The factors of length ≥ 4 of P(a) = aaabb and P(b) = aabbb
457
areaaabb, aabbb, aabb, aaab, abbb. None of them is stronglyτ-irreducible:
458
aaabb, aabbb, aabbareτ-reducible, andaaab, abbbsatisfy one of the forbidden
459
property (1) or (2) of Proposition 7.2. We thus have to introduce a stronger
460
constraint to define a suitableT, which turns out to be
461
T = {anbabn|n >1} (7.1)
7.1. Canonical representative
462
Proposition 7.1. For allP inA1,1 Red∗τ,v(P(τ)) =Red∗τ,v(P(v)).
463
Proof. The proof is by induction on|P|.
464
Basis. IfP =y thenRed∗τ,v(τ) =Red∗τ,v(v) =v.
465
Induction. Let P = uyw and lets =Red∗τ,v(u) ∈Θτ. By Proposition
466
4.6,Red∗τ,v(P(τ)) =Red∗τ,v(sτ w) andRed∗τ,v(P(v)) =Red∗τ,v(svw). Thus, to
467
prove the result it is enough to show thatRedτ,v(sτ w) =svw, i.e., that the
468
shortest prefixsτ of sτ wis sτ. Let us assume that there existss0 such that
469
s0τ is a strict prefix ofsτ. Since sinces∈Θτ,s0τ is not a prefix ofs.
470
s τ
s0 τ
t
471
It follows that there exists a nonempty word t, with 0<|t|<|τ|, which is
472
both a suffix and a prefix ofτ=anbabn, such thats0τ=st.
473
The first letter oft has to bea and its last letterb. Therefore anbis a
474
prefix oftandabnis a suffix oft, hencet=anbabn, contradicting|t|<|τ|.
475
7.2. Strongly irreducible words
476
We state a sufficient condition for a wordw∈ Ato be stronglyτ-irreducible.
477
Proposition 7.2. A nonempty word w is strongly τ-irreducible if it is τ-
478
irreducible and it has the additional properties that τ andwdo not overlap,
479
i.e., there do not exist words u, t0, t such thatt /∈ {ε, τ} and
480
(1) eitherw=ut andτ=tt0,
481
(2) or τ=t0t andw=tu.
482
Proof. It is enough to show that if a factorwoftsatisfies the above hypoth-
483
esis, thenwis a factor ofRedτ,v(t) when|v|<|τ|.
484
Lett=w0τ w00withw0 τ-irreducible. ThenRedτ,v(t) =w0vw00. Aswis
485
τ-irreducible and wand τ do not overlap, if w is a factor oft, it is a factor
486
ofw0 or a factor ofw00, hence a factor ofRedτ,v(t) =w0vw00.
487
The following proposition implies Assumption 4.11.
488
