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The algebra of binary trees is affine complete

Andre Arnold, Patrick Cegielski, Serge Grigorieff, Irene Guessarian

TL;DR

The paper addresses whether congruence-preserving maps on the non-commutative, non-associative algebra of binary trees are necessarily polynomial. It builds a tree-algebra framework $\mathcal{T}(\Sigma)$ with grafting endomorphisms $\gamma_{a\rightarrow \tau}$ and skeleton-based similarity $\sim_s$, and introduces a weakened CP notion (WCP) tied to graftings. The main result shows that for $|\Sigma|\ge 3$, every CP function $f:\mathcal{T}^n\to \mathcal{T}$ equals $\widetilde{P}$ for some polynomial $P$, establishing affine completeness. The proof combines restricting to alphabetic inputs, associating CP with WCP, and an induction on the output size $\sigma(g)$ to reduce to a polynomial form, with potential implications for formal methods in computer science where non-commutative, non-associative algebras arise.

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 polynomial.

The algebra of binary trees is affine complete

TL;DR

The paper addresses whether congruence-preserving maps on the non-commutative, non-associative algebra of binary trees are necessarily polynomial. It builds a tree-algebra framework with grafting endomorphisms and skeleton-based similarity , and introduces a weakened CP notion (WCP) tied to graftings. The main result shows that for , every CP function equals for some polynomial , establishing affine completeness. The proof combines restricting to alphabetic inputs, associating CP with WCP, and an induction on the output size to reduce to a polynomial form, with potential implications for formal methods in computer science where non-commutative, non-associative algebras arise.

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 polynomial.

Paper Structure

This paper contains 8 sections, 15 theorems, 2 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The algebra of binary trees
  3. Trees, congruences
  4. Homomorphisms, graftings
  5. Congruence preserving functions on trees
  6. Equality of CP functions
  7. The algebra of binary trees is affine complete
  8. Conclusion

Key Result

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 t

Figures (1)

  • 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.)

Theorems & Definitions (34)

  • Lemma 2.1
  • Definition 2.2
  • Lemma 2.3: Unicity of decomposition
  • Lemma 2.4
  • Remark 2.5
  • Definition 2.6
  • Definition 2.7: Grafting
  • Proposition 2.8
  • proof
  • Proposition 2.9
  • ...and 24 more