Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A new Boolean matrix representation for Catalan semirings

Mikhail Volkov

Abstract

We construct a faithful representation of the semiring of all order-preserving decreasing transformations of a chain with $n+1$ elements by Boolean upper triangular $n\times n$-matrices.

A new Boolean matrix representation for Catalan semirings

Abstract

We construct a faithful representation of the semiring of all order-preserving decreasing transformations of a chain with elements by Boolean upper triangular -matrices.

Paper Structure

This paper contains 7 sections, 3 theorems, 13 equations.

Table of Contents

  1. Catalan monoids
  2. Catalan semirings
  3. Semirings of Boolean matrices
  4. A faithful representation $\mathcal{C}_n\to\mathcal{T}_n$
  5. A faithful representation $\mathcal{C}^-_{n+1}\to\mathcal{T}_n$
  6. Optimality
  7. Complementarity

Key Result

Theorem 1

The map $\alpha\mapsto M(\alpha)$ is a faithful representation of the semiring $\mathcal{C}^-_{n+1}$ in the semiring $\mathcal{T}^-_n$.

Theorems & Definitions (7)

  • Theorem 1
  • proof
  • Remark
  • Corollary 2
  • Theorem 3
  • proof
  • Remark