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The Layered Catalan Monoids: Structure and Determinants

M. H. Shahzamanian

TL;DR

The paper introduces Layered Catalan Monoids $LC_n$ and studies their semigroup-determinant properties within the $\ll$-smooth framework. It develops a canonical word-form for $LC_n$, proves $LC_n$ is $\ll$-smooth (singleton-rich and $\ll$-transitive), and applies the determinant analysis of Sha-Det2. The main result is that the semigroup determinant $\theta_{LC_n}(X)$ is non-zero precisely for $n \le 7$ and vanishes for $n \ge 8$, with explicit idempotent-decomposition reasoning. The findings illustrate a sharp threshold in determinant behavior for a natural Catalan-inspired monoid class and link to broader algebraic properties and potential coding-theoretic implications.

Abstract

In this paper, we introduce and study a class of monoids, called Layered Catalan Monoids (\( {LC}_n \)), which satisfy the structural conditions for $\ll$-smoothness as defined in~\cite{Sha-Det2}. These monoids are defined by specific identities inspired by Catalan monoids. We establish their canonical forms and compute their determinant, proving that it is non-zero for \(1 \leq n \leq 7\) but vanishes for \(n \geq 8\).

The Layered Catalan Monoids: Structure and Determinants

TL;DR

The paper introduces Layered Catalan Monoids and studies their semigroup-determinant properties within the -smooth framework. It develops a canonical word-form for , proves is -smooth (singleton-rich and -transitive), and applies the determinant analysis of Sha-Det2. The main result is that the semigroup determinant is non-zero precisely for and vanishes for , with explicit idempotent-decomposition reasoning. The findings illustrate a sharp threshold in determinant behavior for a natural Catalan-inspired monoid class and link to broader algebraic properties and potential coding-theoretic implications.

Abstract

In this paper, we introduce and study a class of monoids, called Layered Catalan Monoids (), which satisfy the structural conditions for -smoothness as defined in~\cite{Sha-Det2}. These monoids are defined by specific identities inspired by Catalan monoids. We establish their canonical forms and compute their determinant, proving that it is non-zero for but vanishes for .

Paper Structure

This paper contains 6 sections, 15 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Identities
  4. Determinant of a semigroup
  5. Layered Catalan monoid
  6. Determinant of Layered Catalan monoids

Key Result

Lemma 3.1

In the monoid $LC_n$, where $n \geq 1$, the following identities hold: Note that indices are taken modulo $n$, meaning that if $i+1 > n$ or $i+2 > n$, we interpret them as $(i+1) \bmod n$ and $(i+2) \bmod n$, respectively.

Theorems & Definitions (27)

  • Lemma 3.1
  • proof
  • Proposition 3.2
  • Lemma 3.3
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • Lemma 3.6
  • proof
  • ...and 17 more