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The maximal subsemigroups of the ideals on a monoid of partial injections

Apatsara Sareeto, Jörg Koppitz

Abstract

In the present paper, a submonoid of the well studied monoid $POI_n$ of all order-preserving partial injections on an $n$-element chain is studied. The set $IOF_n^{par}$ of all partial transformations in $POI_n$ which are fence-preserving as well as parity-preserving form a submonoid of $POI_n$. We describe the Green's relations and ideals of $IOF_n^{par}$. For each ideal of $IOF_n^{par}$, we characterize the maximal subsemigroups. We will observe that there are three different types of maximal subsemigroups.

The maximal subsemigroups of the ideals on a monoid of partial injections

Abstract

In the present paper, a submonoid of the well studied monoid of all order-preserving partial injections on an -element chain is studied. The set of all partial transformations in which are fence-preserving as well as parity-preserving form a submonoid of . We describe the Green's relations and ideals of . For each ideal of , we characterize the maximal subsemigroups. We will observe that there are three different types of maximal subsemigroups.
Paper Structure (3 sections, 11 theorems)

This paper contains 3 sections, 11 theorems.

Table of Contents

  1. Introduction and Preliminaries
  2. The ideals
  3. The maximal subsemigroups of the ideals on $IOF_n^{par}$

Key Result

Proposition 1

Apa Let $p\leq n$ and let $\alpha = \bigl(\bigr) \in I_n$. Then $\alpha\in IOF_n^{par}$ if and only if the following four conditions hold: (i) $m_1<m_2< \cdots <m_p$; (ii) $d_1$ and $m_1$ have the same parity; (iii) $d_{i+1}-d_i=1$ if and only if $m_{i+1}-m_i=1$ for all $i\in\{1,...,p-1\}$; (iv) $d_

Theorems & Definitions (21)

  • Proposition 1
  • proof
  • Remark 1
  • Proposition 2
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Lemma 1
  • proof
  • ...and 11 more