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On oriented alternating inverse monoids

Vítor Hugo Fernandes

TL;DR

This work analyzes the oriented and orientation-preserving inverse submonoids $\mathcal{AOR}_n$ and $\mathcal{AOP}_n$ inside the alternating inverse monoid $\mathcal{AI}_n$ on a chain $\Omega_n$ of size $n$. It provides a complete structural and enumerative picture: explicit $\mathscr{J}$-class stratifications, unit groups, and cardinalities, together with a full congruence classification built from Rees-type congruences and subgroup-congruences transferred via carefully constructed lifting maps. The authors deliver concrete, minimal generating sets and exact ranks for both $\mathcal{AOP}_n$ and $\mathcal{AOR}_n$, with results depending on $n$ modulo $4$. These results extend earlier work on orientation-preserving and monotone variants, offering a thorough algebraic framework for oriented and orientation-preserving transformations within the alternating inverse setting. The findings have potential implications for computation and further structural studies of inverse submonoids arising from symmetry and orientation constraints.

Abstract

In this paper, we consider the inverse submonoids $AOR_n$ of oriented transformations and $AOP_n$ of orientation-preserving transformations of the alternating inverse monoid $AI_n$ on a chain with $n$ elements. We compute the cardinalities, describe the Green's structures and the congruences, and calculate the ranks of $AOR_n$ and $AOP_n$.

On oriented alternating inverse monoids

TL;DR

This work analyzes the oriented and orientation-preserving inverse submonoids and inside the alternating inverse monoid on a chain of size . It provides a complete structural and enumerative picture: explicit -class stratifications, unit groups, and cardinalities, together with a full congruence classification built from Rees-type congruences and subgroup-congruences transferred via carefully constructed lifting maps. The authors deliver concrete, minimal generating sets and exact ranks for both and , with results depending on modulo . These results extend earlier work on orientation-preserving and monotone variants, offering a thorough algebraic framework for oriented and orientation-preserving transformations within the alternating inverse setting. The findings have potential implications for computation and further structural studies of inverse submonoids arising from symmetry and orientation constraints.

Abstract

In this paper, we consider the inverse submonoids of oriented transformations and of orientation-preserving transformations of the alternating inverse monoid on a chain with elements. We compute the cardinalities, describe the Green's structures and the congruences, and calculate the ranks of and .

Paper Structure

This paper contains 5 sections, 19 theorems, 50 equations, 2 figures.

Table of Contents

  1. The structure of $\mathcal{AOP}_n$
  2. The structure of $\mathcal{AOR}_n$
  3. The congruences
  4. Generators and rank of $\mathcal{AOP}_n$
  5. Generators and rank of $\mathcal{AOR}_n$

Key Result

Lemma 1.1

Let $\alpha\in J_{n-1}^{\mathcal{I}_n}\cap\mathcal{POI}_n$. Then, $\alpha\in\mathcal{AO}_n$ if and only if $\mathrm{d}(\alpha)$ and $\mathrm{i}(\alpha)$ have the same parity.

Figures (2)

  • Figure 1: The Hasse diagram of $\mathcal{AOP}_n/_\mathscr{J}$, for $n$ even.
  • Figure 2: The Hasse diagram of $\mathcal{AOR}_n/_\mathscr{J}$, for $n\equiv 2\,(\mathrm{mod}{\,4})$.

Theorems & Definitions (19)

  • Lemma 1.1: Fernandes:2024sub
  • Lemma 1.2: Fernandes:2024sub
  • Proposition 1.3
  • Proposition 1.4
  • Lemma 1.5
  • Lemma 1.6
  • Proposition 1.7
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • ...and 9 more