\publicationdetails

232021216890

The algebra of binary trees is affine complete

André Arnold\affiliationmark1 Patrick Cégielski\affiliationmark2 Serge Grigorieff\affiliationmark3 Irène Guessarian\affiliationmark3,4 Université de Bordeaux, France.
LACL, Université Paris XII – IUT de Sénart-Fontainebleau, France.
IRIF, CNRS & Université Paris-Diderot, France.
Emerita Sorbonne Université, Paris

(2020-11-10; 2021-03-08; 2021-05-11)

Abstract

A function on an algebra is congruence preserving if, for any congruence, it maps pairs of congruent elements onto pairs of congruent elements. We show that on the algebra of binary trees whose leaves are labeled by letters of an alphabet containing at least three letters, a function is congruence preserving if and only if it is a polynomial function, thus exhibiting the first example of a non commutative and non associative affine complete algebra.

keywords:

algebras, trees, congruences

Merci à Maurice pour nombre de discussions algébriques passionnantes

1 Introduction

A function on an algebra is congruence preserving if, for any congruence, it maps pairs of congruent elements onto pairs of congruent elements.

A polynomial function on an algebra is a function defined by a term of the algebra using variables, constants and the operations of the algebra. Obviously, every polynomial function is congruence preserving.

Algebras where all congruence preserving functions are polynomial functions are called affine complete in the terminology introduced by Werner (1971). They are extensively studied in the book by Kaarli and Pixley (2001).

In the commutative case, many algebras have been shown to be affine complete: Boolean algebras (Grätzer, 1962), $p$ -rings with unit (Iskander, 1972). For distributive lattices, Ploščica and Haviar (2008) described congruence preserving functions, and Grätzer (1964) determined which distributive lattices are affine complete. Affine completenes is an intrinsic property of an algebra, which fails to hold even for very simple algebras: e.g., in $\mathcal{A}=\langle{\mathbb{Z}},+\rangle$ , the function $f\colon{\mathbb{Z}}\to{\mathbb{Z}}$ defined by

f(x)=\texttt{if $x\geq 0$ then $\dfrac{\Gamma(1/2)}{2\times 4^{x}\times x!}\int_{1}^{\infty}e^{-t/2}(t^{2}-1)^{x}dt$ else $-f(-x)$.}

has been proved to be congruence preserving (Cégielski et al., 2015), but it is not a polynomial function because its power series is infinite. Hence $\mathcal{A}=\langle{\mathbb{Z}},+\rangle$ is not affine complete.

In the non commutative case, very little is known about affine complete algebras. We proved in Arnold et al. (2020) that the free monoid $\Sigma^{*}$ is an associative non commutative affine complete algebra if $\Sigma$ has at least three letters, and we proved in Arnold et al. (2020) a partial result concerning a non commutative and non associative algebra: every unary congruence preserving function $f\colon T(\Sigma)\to T(\Sigma)$ is a polynomial function, where $T(\Sigma)$ is the algebra of full binary trees with leaves labelled by letters of an alphabet $\Sigma$ having at least three letters. We here generalize this result proving that a congruence preserving function $f\colon\mathcal{T}(\Sigma)^{n}\to\mathcal{T}(\Sigma)$ of any arity $n$ is a polynomial function, where $\mathcal{T}(\Sigma)$ is the algebra of arbitrary (possibly non full) binary trees with labelled leaves. This generalization is twofold: (1) non full binary trees are allowed in $\mathcal{T}(\Sigma)$ , and (2) congruence preserving functions of arbitrary arity are allowed. This exhibits an example of a non commutative and non associative affine complete algebra. Non commutative and non associative algebras are of constant use in Computer Science, and congruences are also very often used, whence the potential usefulness of our result.

We first define binary trees and their congruences, we then study conditions which will enable us to prove that every congruence preserving function is a polynomial function, and to finally prove the affine completeness of $T(\Sigma)$ .

2 The algebra of binary trees

2.1 Trees, congruences

For an algebra $\mathcal{A}$ with domain $A$ , a congruence $\sim$ on $\mathcal{A}$ is an equivalence relation on $A$ which is compatible with the operations of $\mathcal{A}$ . We state the characterization of congruences by kernels of homomorphisms.

Lemma 2.1.

Let $\mathcal{A}=\langle A\,,\,\star\rangle$ be an algebra with a binary operation $\star$ . An equivalence $\sim$ on $A$ is a congruence iff there exists an algebra $\mathcal{B}=\langle B\,,\,*\rangle$ with a binary operation $*$ and there exists $\theta\colon A\to B$ a homomorphism such that $\sim$ coincides with the kernel congruence $\ker(\theta)$ of $\theta$ , defined by $x\sim_{\theta}y$ iff $\theta(x)=\theta(y)$ .

Let $\Sigma$ be an alphabet not containing $\{0,1\}$ . We shall represent the algebra of binary trees over $\Sigma$ , i.e., trees with leaves labeled by letters of $\Sigma$ , as a set of words $\mathcal{T}(\Sigma)$ on the alphabet $\Sigma\cup\{0,1\}$ , together with the binary product operation $\star$ .

Definition 2.2.

The algebra $\mathcal{B}=\langle\mathcal{T}(\Sigma),\star\rangle$ of binary trees over $\Sigma$ is defined as follows.

•

A binary tree over $\Sigma$ is a finite set of words $t\subseteq\{0,1\}^{*}\Sigma$ such that: For any $ua,vb\in t$ , if $ua\neq vb$ then $u$ is not a prefix of $v$ and $v$ is not a prefix of $u$ . The carrier set $\mathcal{T}(\Sigma)$ is the set of all binary trees. The empty set $\emptyset$ is a binary tree denoted by ${\bf 0}$ .
•

The binary product operation $\star$ is defined by: for $t,\;t^{\prime}\in\mathcal{T}(\Sigma)$ , $t\star t^{\prime}=0.t\cup 1.t^{\prime}$ . In particular, ${\bf 0}\star{\bf 0}={\bf 0}$ .

Figure 1: From left to right,

t=\{00b,1a\}

\tau=\{0c,1d\}

t_{1}={{\gamma}_{a\rightarrow\tau}}(t)=\{00b,01c,11d\}

t_{2}=\{00b,01c,11d\}

t_{3}=\{00a,10b,11c\}

. Trees

t_{1},\ t_{2},\ t_{3}

have the same size 6, trees

t_{1}

and

t_{3}

are similar (have the same skeleton.)

When the alphabet $\Sigma$ is clear, we will denote by $\mathcal{T}$ the set of all binary trees. Trees are generated by $\{{\bf 0}\}\cup\Sigma$ and the operation $\star$ .

An essential property of this algebra $\mathcal{B}$ is that its elements are uniquely decomposable.

Lemma 2.3 (Unicity of decomposition).

If $t$ is a tree not in $\{{\bf 0}\}\cup\Sigma$ then there exists a unique ordered pair $\langle{t_{1},t_{2}}\rangle\neq\langle{{\bf 0},{\bf 0}}\rangle$ in $\mathcal{T}^{2}$ such that $t=t_{1}\star t_{2}$ .

This property allows us to associate with each $t\in\mathcal{T}$ its size $|t|$ (number of nodes)

– $|{\bf 0}|=0$ , and for all $a\in\Sigma$ , $|a|=1$ ,

– if $t\notin\{{\bf 0}\}\cup\Sigma$ then $t=t_{1}\star t_{2}$ , and $|t|=|t_{1}|+|t_{2}|+1$ .

If $|t|>1$ then there exist $t_{1},t_{2}$ with $|t_{i}|<|t|$ such that $t=t_{1}\star t_{2}$ . Trees $t\star\ t^{\prime}$ , ${\bf 0}\star t^{\prime}$ , $t\star{\bf 0}$ are trees whose root has two sons, a single right son, a single left son, respectively. See Figure 1.

2.2 Homomorphisms, graftings

Lemma 2.4.

Let $\mathcal{B}=\langle B\,,\,*\rangle$ be an algebra with a binary operation $*$ . Every mapping $h\colon\Sigma\to B$ can be uniquely extended to a homomorphism $h\colon\mathcal{T}\to B$ .

Remark 2.5.

1) Because of the universal property of Lemma 2.4, homomorphisms are (uniquely) defined by giving their values on $\Sigma$ .

2) For every endomorphism, $h({\bf 0})={\bf 0}$ . Otherwise, as ${\bf 0}={\bf 0}\star{\bf 0}$ , $h({\bf 0})=h({\bf 0})\star h({\bf 0})$ ; if $h({\bf 0})=t$ with $|t|\geq 1$ then $t=t\star t$ implies $|t|=2|t|+1$ , a contradiction.

Definition 2.6.

For a given $a\in\Sigma$ , let $\nu_{a}$ be the endomorphism sending $\Sigma$ onto $a$ . If for some $a\in\Sigma$ , $\nu_{a}(t)=\nu_{a}(t^{\prime})$ , trees $t$ and $t^{\prime}$ are said to be similar, which is denoted by $t\sim_{s}t^{\prime}$ .

Note that the congruence $\sim_{s}$ does not depend on the choice of the letter $a\in\Sigma$ since $\nu_{b}(t)=\nu_{b}(\nu_{a}(t))$ . From an intuitive viewpoint, $t\sim_{s}t^{\prime}$ means that $t$ and $t^{\prime}$ have the same skeleton, i.e., they are identical except for the leaf labels. See Figure 1.

Other congruences fundamental for our proof are the kernels of the grafting endomorphisms, defined below.

Definition 2.7 (Grafting).

Let $a\in\Sigma$ and $\tau\in\mathcal{T}$ . Then the grafting ${{\gamma}_{a\rightarrow\tau}}\colon\mathcal{T}\to\mathcal{T}$ is the endomorphism defined by its restriction on $\Sigma$

{{\gamma}_{a\rightarrow\tau}}(b)=\begin{cases}\tau&\text{ if }b=a,\\ b&\text{ if }b\neq a.\end{cases}

In other words, for any $a\in\Sigma$ and any $\tau\in\mathcal{T}$ , ${\gamma}_{a\rightarrow\tau}$ is the endomorphism sending the letter $a$ on $\tau$ and each other letter on itself.

An endomorphism $h$ of $\langle\mathcal{T}(\Sigma),\star\rangle$ is idempotent if for every $t\in\mathcal{T}$ , $h(h(t))=h(t)$ . By Lemma 2.4, $h$ is idempotent iff for every $a\in\Sigma$ , $h(h(a))=h(a)$ . For instance if $a$ does not occur in $\tau$ then ${\gamma}_{a\rightarrow\tau}$ is idempotent.

Proposition 2.8.

Let $\tau\in\mathcal{T}$ , let $t,\ t^{\prime}\in\mathcal{T}$ , and let $a_{1}\neq a_{2}$ be two letters in $\Sigma$ . If ${{\gamma}_{a_{i}\rightarrow\tau}}(t)={{\gamma}_{a_{i}\rightarrow\tau}}(t^{\prime})$ for $i=1,2$ , then $t=t^{\prime}$ .

Proof.

By induction on $\min(|t|,|t^{\prime}|)$ .

Basis Case 0: If $\min(|t|,|t^{\prime}|)=0$ then one of $t,t^{\prime}$ is ${\bf 0}$ , say $t={\bf 0}$ . If $t^{\prime}\neq{\bf 0}$ then $t^{\prime}$ contains at least one occurrence of some letter $b$ . As ${\gamma}_{a_{i}\rightarrow\tau}(t^{\prime})={\gamma}_{a_{i}\rightarrow\tau}(t)={\gamma}_{a_{i}\rightarrow\tau}({\bf 0})={\bf 0}$ , we have ${\gamma}_{a_{i}\rightarrow\tau}(t^{\prime})={\bf 0}$ , which implies (because $t^{\prime}\neq{\bf 0}$ was supposed) that $\tau={\bf 0}$ . Then ${\gamma}_{a_{i}\rightarrow\tau}(t^{\prime})={\bf 0}$ implies that all leaves of $t^{\prime}$ are equal to both $a_{1}$ and $a_{2}$ , a contradiction. Hence $t^{\prime}={\bf 0}$ and $t=t^{\prime}$ .

Basis Case 1: If $\min(|t|,|t^{\prime}|)=1$ then $t$ or $t^{\prime}$ is a letter, say $t=b$ , and there is one $i$ , say $i=1$ , such that $a_{1}\neq b$ , thus $b={{\gamma}_{a_{1}\rightarrow\tau}}(t)={{\gamma}_{a_{1}\rightarrow\tau}}(t^{\prime})$ .

•

If $t^{\prime}$ is a letter $c\neq b$ , then ${{\gamma}_{a_{1}\rightarrow\tau}}(c)=b$ . If $c=a_{1}$ then $b={{\gamma}_{a_{1}\rightarrow\tau}}(c)=\tau$ . Since ${\gamma}_{a_{2}\rightarrow\tau}(c)=c={\gamma}_{a_{2}\rightarrow\tau}(b)\in\{\tau,b\}=\{b\}$ , we have that $c=b$ , a contradiction. If $c\neq a_{1}$ and ${{\gamma}_{a_{1}\rightarrow\tau}}(c)=c\neq b={{\gamma}_{a_{1}\rightarrow\tau}}(c)$ , a contradiction. Hence $t^{\prime}=t=b$ .
•

If $|t^{\prime}|>1$ then $t^{\prime}=t^{\prime}_{1}\star t^{\prime}_{2}$ , and ${{\gamma}_{a_{1}\rightarrow\tau}(t^{\prime})}={{\gamma}_{a_{1}\rightarrow\tau}(t^{\prime}_{1})}\star{{\gamma}_{a_{1}\rightarrow\tau}(t^{\prime}_{2})}$ which can be only of size 0 or $\geq 2$ , contradicting ${{\gamma}_{a_{1}\rightarrow\tau}}(t^{\prime})=b$ . this case is excluded.

Induction: If $\min(|t|,|t^{\prime}|)>1$ then $t=t_{1}\star t_{2}$ and $t^{\prime}=t^{\prime}_{1}\star t^{\prime}_{2}$ with $\min(|t_{i}|,|t^{\prime}_{i}|)<\min(|t|,|t^{\prime}|)$ , for $i=1,2$ . By Lemma 2.3, ${{\gamma}_{a_{j}\rightarrow\tau}}(t_{1})\star{{\gamma}_{a_{j}\rightarrow\tau}}(t_{2})={{\gamma}_{a_{j}\rightarrow\tau}}(t^{\prime}_{1})\star{{\gamma}_{a_{j}\rightarrow\tau}}(t^{\prime}_{2})$ implies ${{\gamma}_{a_{j}\rightarrow\tau}}(t_{i})={{\gamma}_{a_{j}\rightarrow\tau}}(t^{\prime}_{i})$ , for $j=1,2$ . By the induction hypothesis $t_{i}=t^{\prime}_{i}$ , hence $t=t^{\prime}$ . ∎

Proposition 2.9.

Let us fix $a\in\Sigma$ , with $|\Sigma|\geq 3$ , $t,\ t^{\prime}\in\mathcal{T}$ such that $t\sim_{s}t^{\prime}$ .

(1) If, for some $\tau\in\mathcal{T}$ of size $|\tau|\neq 1$ , ${{\gamma}_{a\rightarrow\tau}}(t)={{\gamma}_{a\rightarrow\tau}}(t^{\prime})$ , then $t=t^{\prime}$ .

(2) If, for all $b\neq a$ , $b\in\Sigma$ , ${{\gamma}_{a\rightarrow b}}(t)={{\gamma}_{a\rightarrow b}}(t^{\prime})$ , then $t=t^{\prime}$ .

Proof.

Both (1) and (2) are proved by induction on $|t|=|t^{\prime}|$ , and in both cases, the result obviously holds if $t=t^{\prime}={\bf 0}$ .

Basis: If $|t|=|t^{\prime}|=1$ .

(1) We assume that $t=b\neq c=t^{\prime}$ .

(i) If $a\not\in\{b,c\}$ then ${{\gamma}_{a\rightarrow\tau}}(t)=b\neq c={{\gamma}_{a\rightarrow\tau}}(t^{\prime})$ , a contradiction.

(ii) Otherwise, $a\in\{b,c\}$ , e.g., $a=b=t$ , then ${{\gamma}_{a\rightarrow\tau}}(t)={{\gamma}_{a\rightarrow\tau}}(a)=\tau$ and ${{\gamma}_{a\rightarrow\tau}}(t^{\prime})={{\gamma}_{a\rightarrow\tau}}(c)=c$ , hence $\tau=c$ , which contradicts $|\tau|\neq 1$ .

(2) We assume that $t=b\neq c=t^{\prime}$ .

(i) The case $a\not\in\{b,c\}$ yields a contradiction as in case (1).

(ii) Otherwise, e.g., $a=b$ , there exists $d\not\in\{a,c\}$ , and we get ${{\gamma}_{a\rightarrow d}}(t)={{\gamma}_{a\rightarrow d}}(a)=d$ and ${{\gamma}_{a\rightarrow d}}(t^{\prime})={{\gamma}_{a\rightarrow d}}(c)=c$ , contradicting ${{\gamma}_{a\rightarrow d}}(t)={{\gamma}_{a\rightarrow d}}(t^{\prime})$ .

Induction: As in Proposition 2.8 in both cases: since $t$ and $t^{\prime}$ are similar, $t=t_{1}\star t_{2}$ and $t^{\prime}=t^{\prime}_{1}\star t^{\prime}_{2}$ with $t_{i}$ similar to $t^{\prime}_{i}$ and $|t_{i}|<|t^{\prime}_{i}|$ . ∎

2.3 Congruence preserving functions on trees

Definition 2.10.

A function $f\colon\mathcal{T}^{n}\to\mathcal{T}$ is congruence preserving (abbreviated into CP) if for all congruences $\sim$ on $\mathcal{T}$ , for all $t_{1},\ldots,t_{n},\ t^{\prime}_{1},\ldots,t^{\prime}_{n}$ in $\mathcal{T}$ , $t_{i}\sim t^{\prime}_{i}$ for all $i=1,\ldots,n$ , implies $f(t_{1},\ldots,t_{n})\sim f(t^{\prime}_{1},\ldots,t^{\prime}_{n})$ .

Remark 2.11.

(1) It follows from Lemma 2.1 that CP functions are characterized by the fact that for all homomorphisms $h$ from $\langle{{\mathcal{T}},\star}\rangle$ to any algebra $\langle{A,\star_{A}}\rangle$ , $h(t_{i})=h(t^{\prime}_{i})$ for all $i=1,\ldots,n$ , implies $h(f(t_{1},\ldots,t_{n}))=h(f(t^{\prime}_{1},\ldots,t^{\prime}_{n}))$ .

(2) If $f$ is CP and endomorphism $h$ is idempotent then $h(f(t_{1},\ldots,t_{n}))=h(f(h(t_{1}),\ldots,h(t_{n})))$ . Indeed, let $\sim_{h}$ be the congruence associated with $h$ , for $i=1,\ldots,n$ , we have $h(t_{i})=h(h(t_{i}))$ , hence $t_{i}\sim_{h}h(t_{i})$ , whence the result.

We will show that congruence preserving functions on the algebra $\langle\mathcal{T}(\Sigma),\star\rangle$ are polynomial functions. Let us first formally define polynomials on trees.

Definition 2.12.

Let $x_{1},\ldots,x_{n}\not\in\Sigma$ be called variables. A polynomial $P(x_{1},\ldots,x_{n})$ is a tree on the alphabet $\Sigma\cup\{x_{1},\ldots,x_{n}\}$ .

With every polynomial $P(x_{1},\ldots,x_{n})$ we will associate a polynomial function $\tilde{P}\colon\mathcal{T}^{n}\to\mathcal{T}$ defined by: for any $\vec{u}=\langle t_{1},\ldots,t_{i},\ldots,t_{n}\rangle\in\mathcal{T}^{n}$ ,

$\tilde{P}(\vec{u})=\left\{\begin{array}[]{ll}P&\mbox{if $P={\bf 0}$ or $P\in\Sigma$}\\ t_{i}&\mbox{if $P=x_{i}$}\\ \widetilde{P_{1}}(\vec{u})\star\widetilde{P_{2}}(\vec{u})&\mbox{if $P=P_{1}\star P_{2}$}\end{array}\right.$

Obviously, every polynomial function is CP. Our goal is to prove the converse, namely

Theorem 2.13.

Let $|\Sigma|\geq 3$ . If $g\colon\mathcal{T}^{n}\to\mathcal{T}$ is CP then there exists a polynomial $P_{g}$ such that $g=\widetilde{P_{g}}$ .

3 Equality of CP functions

Notation 3.1.

For any $f\colon{\mathcal{T}}^{n}\to{\mathcal{T}}$ , we denote by $f\raise-2.15277pt\hbox{$|$}_{\Sigma^{n}}$ its restriction to $\Sigma^{n}$ .

In this section we prove that if $f$ and $g$ are two CP functions, then $f\raise-2.15277pt\hbox{$|$}_{\Sigma^{n}}=g\raise-2.15277pt\hbox{$|$}_{\Sigma^{n}}$ implies $f=g$ , provided that $\Sigma$ contains at least three letters.

Lemma 3.2.

Suppose $\Sigma$ has at least three letters. If $f$ and $g$ are unary CP functions on $\mathcal{T}$ such that for all $a\in\Sigma$ , $f(a)=g(a)$ then for all $t\in\mathcal{T}$ , $f(t)$ and $g(t)$ are similar.

Proof.

We have to show that $\nu_{a}(f(t))=\nu_{a}(g(t))$ for some $a\in\Sigma$ and for all $t$ . As $\nu_{a}$ is idempotent and $f$ is CP, by Remark 2.11 (2), $\nu_{a}(f(t))=\nu_{a}(f(\nu_{a}(t)))$ , and similarly for $g$ . Hence it suffices to prove $f(\nu_{a}(t))=g(\nu_{a}(t))$ . Let $b_{1},\ b_{2}\in\Sigma$ such that $a,\ b_{1},\ b_{2}$ are pairwise distinct. As ${\gamma}_{b_{i}\rightarrow\nu_{a}(t)}$ is idempotent, by Remark 2.11 (2), we have ${\gamma}_{b_{i}\rightarrow\nu_{a}(t)}(f(b_{i}))={\gamma}_{b_{i}\rightarrow\nu_{a}(t)}(f(\nu_{a}(t)))$ . The same holds for $g$ , i.e., ${\gamma}_{b_{i}\rightarrow\nu_{a}(t)}(g(b_{i}))={\gamma}_{b_{i}\rightarrow\nu_{a}(t)}(g(\nu_{a}(t)))$ . From $f(b_{i})=g(b_{i})$ , we deduce that ${\gamma}_{b_{i}\rightarrow\nu_{a}(t)}(f(\nu_{a}(t)))={\gamma}_{b_{i}\rightarrow\nu_{a}(t)}(g(\nu_{a}(t)))$ . This equality holds for $i=1,2$ , thus Proposition 2.8 implies that $f(\nu_{a}(t))=g(\nu_{a}(t))$ . ∎

The following proposition shows that a unary CP function $f$ is completely determined by its values on $\Sigma$ .

Proposition 3.3.

Suppose $\Sigma$ has at least three letters. If $f$ and $g$ are unary CP functions on $\mathcal{T}$ such that for all $a\in\Sigma$ , $f(a)=g(a)$ then for all $t\in\mathcal{T}$ , $f(t)=g(t)$ .

Proof.

Let $a$ be a letter that occurs in $t$ . For any other letter $b$ , the endomorphisms ${\gamma}_{a\rightarrow b}$ and ${\gamma}_{a\rightarrow t_{b}}$ are idempotent, where $t_{b}={\gamma}_{a\rightarrow b}(t)$ . Thus by Remark 2.11 (2), ${\gamma}_{a\rightarrow t_{b}}(f(a))={\gamma}_{a\rightarrow t_{b}}(f(t_{b}))$ , and ${\gamma}_{a\rightarrow t_{b}}(g(a))={\gamma}_{a\rightarrow t_{b}}(g(t_{b}))$ . As $f(a)=g(a)$ we have ${\gamma}_{a\rightarrow t_{b}}(f(t_{b}))={\gamma}_{a\rightarrow t_{b}}(g(t_{b}))$ . By Lemma 3.2, $f(t_{b})$ and $g(t_{b})$ are similar, and by Proposition 2.9 (1) $f(t_{b})=g(t_{b})$ .

On the other hand, as $f$ and $g$ are CP and $t\sim_{{\gamma}_{a\rightarrow b}}t_{b}$ , we get ${\gamma}_{a\rightarrow b}(f(t))={\gamma}_{a\rightarrow b}(f(t_{b}))$ and ${\gamma}_{a\rightarrow b}(g(t))={\gamma}_{a\rightarrow b}(g(t_{b}))$ , hence ${\gamma}_{a\rightarrow b}(f(t))={\gamma}_{a\rightarrow b}(g(t))$ . As this is true for all $b\neq a$ , we have by Proposition 2.9 (2), $f(t)=g(t)$ . ∎

Proposition 3.3 now can be generalized.

Notation 3.4.

For any function $f\colon\mathcal{T}^{n+1}\to\mathcal{T}$ , any $t\in\mathcal{T}$ , and $\vec{u}=\langle t_{1},\ldots,t_{n}\rangle$ , we define

(1) a $n$ -ary function $f_{\cdots,t}$ obtained by “freezing” the (n+1)th argument to the value $t$ , and defined by: for all $\vec{u}\in\mathcal{T}^{n}$ , $f_{\cdots,t}(\vec{u})=f(\vec{u},t)$ ,

(2) a unary function $f_{\vec{u},\cdot}$ obtained by “freezing” the $n$ first arguments to the value $\vec{u}=\langle t_{1},\ldots,t_{n}\rangle$ , and defined by: for all $t\in\mathcal{T}$ , $f_{\vec{u},\cdot}(t)=f(\vec{u},t)$ .

Proposition 3.5.

Let $f$ and $g$ be n-ary CP functions on $\mathcal{T}$ such that for all $a_{1},\ldots,a_{n}\in\Sigma$ , $f(a_{1},\ldots,a_{n})=g(a_{1},\ldots,a_{n})$ then for all $t_{1},\ldots,t_{n}\in\mathcal{T}$ , $f(t_{1},\ldots,t_{n})=g(t_{1},\ldots,t_{n})$ .

Proof.

By induction on $n$ . For $n=1$ the result was proved in Proposition 3.3. Assume the result holds for $n$ . By the hypothesis, for all $a_{1},\ldots,a_{n},a\in\Sigma$ , we have $f(a_{1},\ldots,a_{n},a)=g(a_{1},\ldots,a_{n},a)$ , i.e., $f_{\cdots,a}(a_{1},\ldots,a_{n})=g_{\cdots,a}(a_{1},\ldots,a_{n})$ . By the induction applied to $f_{\cdots,a}$ , for all $\vec{u}\in\mathcal{T}^{n}$ , $f_{\cdots,a}(\vec{u})=g_{\cdots,a}(\vec{u})$ , or equivalently $f_{\vec{u},\cdot}(a)=g_{\vec{u},\cdot}(a)$ . As $f_{\vec{u},\cdot}(a)=g_{\vec{u},\cdot}(a)$ , applying now Proposition 3.3 to $f_{\vec{u},\cdot}$ and $g_{\vec{u},\cdot}$ yields $f_{\vec{u},\cdot}(t)=g_{\vec{u},\cdot}(t)$ for all $t$ , hence $f(\vec{u},t)=g(\vec{u},t)$ . ∎

4 The algebra of binary trees is affine complete

To prove that any CP function is a polynomial function, as a consequence of Proposition 3.5 and of the fact that a polynomial function is CP, it is enough to show that the restriction $f\raise-2.15277pt\hbox{$|$}_{\Sigma^{n}}$ of $f\colon{\mathcal{T}}^{n}\to{\mathcal{T}}$ to $\Sigma^{n}$ is equal to the restriction $\tilde{P}\raise-2.15277pt\hbox{$|$}_{\Sigma^{n}}$ of a $n$ -ary polynomial function. For such restricted functions we introduce a weakened version WCP of the CP condition, namely,

Definition 4.1.

Function $g\colon\mathcal{T}^{n}\to\mathcal{T}$ is said to be WCP iff for any idempotent mapping $h\colon\Sigma\to\Sigma$ , $\forall\vec{u},\vec{v}\in\Sigma^{n}$ , $h(\vec{u})=h(\vec{v})\implies h(g(\vec{u}))=h(g(\vec{v}))$ , where for $\vec{u}=\langle u_{1},\ldots,u_{n}\rangle$ , $h(\vec{u})$ denotes $\langle h(u_{1}),\ldots,h(u_{n})\rangle$ .

Every CP function is clearly WCP.

Lemma 4.2.

If $g$ is WCP then for all $\vec{u},\vec{v}\in\Sigma^{n}$ , $g(\vec{u})$ and $g(\vec{v})$ are similar.

Proof.

As $\nu_{a}(\vec{u})=\nu_{a}(\vec{v})=\langle a,\ldots,a\rangle$ for all $\vec{u},\vec{v}\in\Sigma^{n}$ and $g$ is WCP, $\nu_{a}(g(\vec{u}))=\nu_{a}(g(\vec{v}))$ . ∎

We often use a different form of the condition WCP, which deals only with alphabetic graftings.

Proposition 4.3.

A function $g$ is WCP if and only if

(GCP) (G for graftings) for all $a_{1},a_{2},\ldots,a_{n}\in\Sigma$ , $i\in\{1,\ldots,n\}$ and $b_{i}\in\Sigma$ , ${\gamma}_{a_{i}\rightarrow b_{i}}(g(a_{1},\ldots,a_{n}))={\gamma}_{a_{i}\rightarrow b_{i}}(g(a_{1},\ldots,a_{i-1},b_{i},a_{i+1},\ldots,a_{n}))$ .

Proof.

Since ${\gamma}_{a_{i}\rightarrow b_{i}}(a_{1},\ldots,a_{n})={\gamma}_{a_{i}\rightarrow b_{i}}(a_{1},\ldots,a_{i-1},b_{i},a_{i+1},\ldots,a_{n})$ , clearly WCP implies GCP. The proof of the converse is by induction on $n$ . It is obviously true for $n=0$ .

Otherwise, let $h$ be a mapping $h\colon\Sigma\to\Sigma$ and let $\vec{u},\vec{v}\in\Sigma^{n}$ such that $h(\vec{u})=h(\vec{v})$ , and let $a,b\in\Sigma$ such that $h(a)=h(b)$ . By (GCP), we have ${\gamma}_{a\rightarrow b}(g(\vec{u},a))={\gamma}_{a\rightarrow b}(g(\vec{u},b))$ , hence $h({\gamma}_{a\rightarrow b}(g(\vec{u},a)))=h({\gamma}_{a\rightarrow b}(g(\vec{u},b)))$ .

But $h({\gamma}_{a\rightarrow b}(c))=\left\{\begin{array}[]{ll}h(c)&\mbox{if $c\neq a$}\\ h(b)=h(a)&\mbox{if $c=a$}\end{array}\right.$ hence $h\circ{\gamma}_{a\rightarrow b}=h$ . Therefore $h(g(\vec{u},a))=h(g(\vec{u},b))$ , and by the induction applied to $g_{\ldots,b}$ , $h(g(\vec{u},a))=h(g(\vec{u},b))=h(g(\vec{v},b))$ . ∎

Let us first study unary WCP functions whose restriction to $\Sigma$ takes its values in $\Sigma$ .

Proposition 4.4.

Assume $|\Sigma|\geq 3$ . Let $f\colon\mathcal{T}\to\mathcal{T}$ be WCP and such that $f(\Sigma)\subseteq\Sigma$ . Then $f\raise-2.15277pt\hbox{$|$}_{\Sigma}$ is (1) either a constant function (2) or the identity.

Proof.

If $f$ is not the identity there exist $a,\ b$ , with $a\neq b$ and $f(a)=b$ . As ${\gamma}_{a\rightarrow b}(f(b))={\gamma}_{a\rightarrow b}(f(a))={\gamma}_{a\rightarrow b}(b)=b$ , we get $f(b)\in\{a,b\}$ .

For $c\not\in\{a,b\}$ , ${\gamma}_{a\rightarrow c}(f(c))={\gamma}_{a\rightarrow c}(f(a))=b$ implies $f(c)=b$ . It remains to prove that $f(b)=b$ . From ${\gamma}_{b\rightarrow c}(f(b))={\gamma}_{b\rightarrow c}(f(c))=c$ , we deduce that $f(b)\in\{c,b\}$ , hence $f(b)\in\{a,b\}\cap\{c,b\}=\{b\}$ , which concludes the proof. ∎

We now will generalize Proposition 4.4 by Proposition 4.5 (replacing a unary $f$ by a $n$ -ary $g$ ).

Proposition 4.5.

Assume $|\Sigma|\geq 3$ . If $g\colon\mathcal{T}^{n}\to\mathcal{T}$ is WCP and such that $g(\Sigma^{n})\subseteq\Sigma$ , then $g\raise-2.15277pt\hbox{$|$}_{\Sigma^{n}}$ is (1) either a constant function (2) or a projection $\pi^{n}_{i}$ .

Proof.

The proof is by induction on $n$ . By Proposition 4.4 it is true for $n=1$ . If $g$ is of arity $n+1$ then, by induction hypothesis, for any $a\in\Sigma$ , the function $g_{\cdots,a}$ of arity $n$ is either a constant or a projection $\pi_{i}^{n}$ . We first show that these functions are all equal to a given $\pi_{i}^{n}$ , or all equal to a same constant, or every $g_{\cdots,a}$ is the constant function $a$ .

Let us assume that $g_{\cdots,a}=\pi_{i}^{n}$ . Let $\vec{u}=\langle{a,\ldots,a,c,a,\ldots,a}\rangle$ and $\vec{v}=\langle{a,\ldots,a,d,a,\ldots,a}\rangle$ with $a,c,d$ pairwise disjoint, so that for any $b$ , ${\gamma}_{a\rightarrow b}(g(\vec{u},a))=c$ and ${\gamma}_{a\rightarrow b}(g(\vec{v},a))=d$ . It follows from the GCP condition that ${\gamma}_{a\rightarrow b}(g(\vec{u},a))={\gamma}_{a\rightarrow b}(g(\vec{u},b))=c$ and ${\gamma}_{a\rightarrow b}(g(\vec{v},a))={\gamma}_{a\rightarrow b}(g(\vec{v},b))=d$ , which is impossible if $g_{\cdots,b}$ is either a constant or a projection $\pi_{j}^{n}$ with $j\neq i$ . Hence all $g_{\cdots,a}$ are equal to $\pi_{i}^{n}$ , implying $g=\pi_{i}^{n+1}$ .

Assume now all the $g_{\cdots,a}$ are constant. For every $\vec{u},\vec{v},a$ , we have $g(\vec{u},a)=g(\vec{v},a)$ . We choose an arbitrary $\vec{u}\in\Sigma^{n}$ which will be fixed. By the induction hypothesis $g_{\vec{u},\cdot}$ is either (1) the identity, or (2) a constant $c$ . In case (1), for all $\vec{v},a$ , $g(\vec{u},a)=g(\vec{v},a)=a$ and $g=\pi_{n+1}^{n+1}$ . In case (2), for all $\vec{v},a,b$ , $g(\vec{u},a)=g(\vec{v},b)=c$ and $g$ is a constant. ∎

As CP functions are WCP, for $g$ a CP function such that for some $a_{1},\ldots,a_{n}\in\Sigma$ , $g(a_{1},\ldots,a_{n})\in\Sigma$ , we have shown that there exists a polynomial $P_{g}$ , which is either a constant $a\in\Sigma$ or an $x_{i}$ , such that $g=\widetilde{P_{g}}$ . We will now extend to the case when $g(a_{1},\ldots,a_{n})\not\in\Sigma$ .

Proposition 4.6.

Assume that $|\Sigma|\geq 3$ . Let $g\colon\mathcal{T}^{n}\to\mathcal{T}$ be WCP. Then there exists a polynomial $P_{g}$ such that $g\raise-2.15277pt\hbox{$|$}_{\Sigma^{n}}=\widetilde{P_{g}}\raise-2.15277pt\hbox{$|$}_{\Sigma^{n}}$ .

Proof.

Let $\sigma(g)$ be the common size of all the $g(\vec{u}),\ \vec{u}\in\Sigma^{n}$ . The proof is by induction on $\sigma(g)$ .

Basis: If $\sigma(g)=0$ then $g\raise-2.15277pt\hbox{$|$}_{\Sigma^{n}}=\tilde{P}\raise-2.15277pt\hbox{$|$}_{\Sigma^{n}}={\bf 0}$ . If $\sigma(g)=1$ then $g(a_{1},\ldots,a_{n})\in\Sigma$ and the result is proved in Proposition 4.5.

Induction: If $\sigma(g)>1$ there exists two functions $g_{i}\colon{\mathcal{T}}^{n}\to{\mathcal{T}}$ for $i=1,2$ such that for all $\vec{u}\in\Sigma^{n}$ , $g(\vec{u})=g_{1}(\vec{u})\star g_{2}(\vec{u})$ , with $|\sigma(g_{i})|<|\sigma(g)|$ . It remains to show that both $g_{1}$ and $g_{2}$ are WCP. Let $\vec{u},\vec{v}\in\Sigma^{n}$ be such that $h(\vec{u})=h(\vec{v})$ for some mapping $h\colon\Sigma\to\Sigma$ . Extend $h$ as an endomorphism $\mathcal{T}\to\mathcal{T}$ by Lemma 2.4, then $h(g(\vec{u}))=h(g_{1}(\vec{u})\star g_{2}(\vec{u}))=h(g_{1}(\vec{u}))\star h(g_{2}(\vec{u}))$ . Similarly, $h(g(\vec{v}))=h(g_{1}(\vec{v}))\star h(g_{2}(\vec{v}))$ . As $g$ is WCP and $h(\vec{u})=h(\vec{v})$ , we have $h(g(\vec{u}))=h(g(\vec{v}))$ . Then by Lemma 2.3 (unique decomposition) we get $h(g_{i}(\vec{u}))=h(g_{i}(\vec{v}))$ for $i=1,2$ . This is true for any $h$ , thus $g_{1}$ and $g_{2}$ are WCP. By the induction hypothesis there exists $P_{i}$ such $\widetilde{P_{i}}\raise-2.15277pt\hbox{$|$}_{\Sigma^{n}}={g_{i}}\raise-2.15277pt\hbox{$|$}_{\Sigma^{n}}$ , hence ${g}\raise-2.15277pt\hbox{$|$}_{\Sigma^{n}}=\widetilde{P_{1}}\raise-2.15277pt\hbox{$|$}_{\Sigma^{n}}\star\widetilde{P_{2}}\raise-2.15277pt\hbox{$|$}_{\Sigma^{n}}=\widetilde{P_{1}\star P_{2}}\raise-2.15277pt\hbox{$|$}_{\Sigma^{n}}$ . ∎

Theorem 4.7.

If $f\colon{\mathcal{T}}^{n}\to{\mathcal{T}}$ is CP then there exists a polynomial $P$ such that $f=\tilde{P}$ .

Proof.

Since $f$ is CP, $f$ also is WCP. By the previous proposition, there exists $P$ such that $f\raise-2.15277pt\hbox{$|$}_{\Sigma^{n}}=\tilde{P}\raise-2.15277pt\hbox{$|$}_{\Sigma^{n}}$ , and by Proposition 3.5, $f=\tilde{P}$ . ∎

5 Conclusion

We proved that, when $\Sigma$ has at least three letters, the algebra of arbitrary binary trees with leaves labeled by letters of $\Sigma$ is an affine complete algebra (non commutative and non associative).

Acknowledgements.

We thanks the referees for comments which helped to improve the paper.

References

Arnold et al. (2020) Arnold A., Cégielski P., Grigorieff S., Guessarian I.: Affine completeness of the algebra of full binary trees. Algebra Universalis, Springer Verlag, 81, Article 55, https://doi.org/10.1007/s00012-020-00690-681:55 (2020)
Cégielski et al. (2015) Cégielski, P., Grigorieff S., Guessarian I.: Newton representation of functions over natural integers having integral difference ratios. Int. J. Number Theory, 11, 2109 – 2139 (2015)
Grätzer (1962) Grätzer G.: On Boolean functions (notes on lattice theory. II). Rev. Math. Pures Appl. (Académie de la République Populaire Roumaine) 7, 693 – 697 (1962)
Grätzer (1964) Grätzer G.: Boolean functions on distributive lattices. Acta Math. Hungar. 15,193 – 201 (1964)
Iskander (1972) Iskander A.: Algebraic functions on $p$ -rings. Colloq. Math. 25, 37 – 41 (1972)
Kaarli and Pixley (2001) Kaarli K., Pixley A.F.: Polynomial Completeness in Algebraic Systems. Chapman & Hall/CRC (2001)
Ploščica and Haviar (2008) Ploščica M., Haviar M.: Congruence-preserving functions on distributive lattices. Algebra Universalis, 59, 179 – 196 (2008)
Werner (1971) Werner H.: Produkte von KongruenzenKlassengeometrien universeller Algebren. Math. Z. 121, 111 – 140 (1971)