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Commuting upper triangular binary morphisms

Juha Honkala

Abstract

A morphism $g$ from the free monoid $X^*$ into itself is called upper triangular if the matrix of $g$ is upper triangular. We characterize all upper triangular binary morphisms $g_1$ and $g_2$ such that $g_1g_2=g_2g_1$.

Commuting upper triangular binary morphisms

Abstract

A morphism from the free monoid into itself is called upper triangular if the matrix of is upper triangular. We characterize all upper triangular binary morphisms and such that .
Paper Structure (11 sections, 14 theorems, 41 equations)

This paper contains 11 sections, 14 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Definitions
  3. Connections between freeness and commutativity
  4. Examples
  5. Properties of infinite words generated by upper triangular binary morphisms
  6. Commuting nonsingular morphisms $g_1$ and $g_2$ such that both $g_1(b)$ and $g_2(b)$ have at least two occurrences of $b$
  7. The numbers of occurrences of $b$ are multiplicatively independent
  8. The numbers of occurrences of $b$ are multiplicatively dependent
  9. Commuting nonsingular morphisms $g_1$ and $g_2$ such that $|g_1(b)|_b=1$ and $|g_2(b)|_b\geq 2$
  10. Commuting nonsingular morphisms $g_1$ and $g_2$ such that $|g_1(b)|_b=|g_2(b)|_b=1$
  11. Commuting morphisms $g_1$ and $g_2$ such that $g_1$ or $g_2$ is singular

Key Result

Theorem 3.1

Let $X=\{a,b\}$ and let $g_1,g_2\in \mathop{\mathrm{Tri}}\nolimits(X^*)$ be nonsingular upper triangular morphisms. Assume that $g_1\neq g_2$. Assume that $\{g_1,g_2\}$ is not a special pair. If $\{g_1,g_2\}$ is not free, then $g_1g_2=g_2g_1$.

Theorems & Definitions (21)

  • Theorem 3.1
  • Theorem 3.2
  • Lemma 3.3
  • Example 4.1
  • Example 4.2
  • Example 4.3
  • Example 4.4
  • Example 4.5
  • Example 4.6
  • Example 4.7
  • ...and 11 more