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On monotone alternating inverse monoids

Vítor Hugo Fernandes

TL;DR

This paper analyzes two inverse submonoids of the alternating inverse monoid $ ext{AI}_n$: the monotone submonoid $ ext{AM}_n$ and the order-preserving submonoid $ ext{AO}_n$. It provides a detailed description of their Green's structures, characterizes all congruences (with $ ext{AO}_n$ admitting exactly $n+3$ Rees congruences and $ ext{AM}_n$ exhibiting a mod-$4$-dependent congruence lattice), and determines their ranks along with explicit generating sets. The results yield precise counts of elements, a clear stratification by $ ext{J}$-classes (including two top rank-$n-1$ classes in several cases), and concrete minimal generating sets whose sizes depend on $nmod 4$. These findings advance understanding of monotone and order-preserving submonoids within alternating inverse semigroups and provide tools for further algebraic and computational exploration.

Abstract

In this paper, we consider the inverse submonoids $AM_n$ of monotone transformations and $AO_n$ of order-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 these two submonoids of $AI_n$.

On monotone alternating inverse monoids

TL;DR

This paper analyzes two inverse submonoids of the alternating inverse monoid : the monotone submonoid and the order-preserving submonoid . It provides a detailed description of their Green's structures, characterizes all congruences (with admitting exactly Rees congruences and exhibiting a mod--dependent congruence lattice), and determines their ranks along with explicit generating sets. The results yield precise counts of elements, a clear stratification by -classes (including two top rank- classes in several cases), and concrete minimal generating sets whose sizes depend on . These findings advance understanding of monotone and order-preserving submonoids within alternating inverse semigroups and provide tools for further algebraic and computational exploration.

Abstract

In this paper, we consider the inverse submonoids of monotone transformations and of order-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 these two submonoids of .

Paper Structure

This paper contains 5 sections, 30 theorems, 28 equations, 4 figures.

Table of Contents

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

Key Result

Proposition 1.1

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

Figures (4)

  • Figure 1: The Hasse diagram of $\mathcal{AO}_n/_\mathscr{J}$.
  • Figure 2: The Hasse diagram of $\mathcal{AM}_n/_\mathscr{J}$, for $n\equiv \{1,2\}\,(\mathrm{mod}{\,4})$.
  • Figure 3: The Hasse diagram of $\mathop{\mathrm{Con}}\nolimits(\mathcal{AO}_n)$.
  • Figure 4: The Hasse diagram of $[\sim_{\hbox{$\!_{F_{n-2}}$}},\sim_{\hbox{$\!_{F_{n-1}}$}}]$, for $n\equiv \{1,2\}\,(\mathrm{mod}{\,4})$.

Theorems & Definitions (30)

  • Proposition 1.1
  • Proposition 1.2
  • Proposition 1.3
  • Lemma 2.1
  • Proposition 2.2
  • Corollary 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Lemma 3.1
  • Lemma 3.2
  • ...and 20 more