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On certain semigroups of finite oriented and order-decreasing partial transformations

Gonca Ayık, Hayrullah Ayık, Ilinka Dimitrova, Jörg Koppitz

Abstract

Let $\mathcal{PORD}_{n}$ be the semigroup consisting of all oriented and order-decreasing partial transformations on the finite chain $X_{n}=\{ 1<\cdots<n \}$. Let $\mathcal{IORD}_{n}$ be the subsemigroup of $\mathcal{PORD}_{n}$ consisting of all injective transformations of $\mathcal{PORD}_{n}$. For $2\leq r\leq n$, let $\mathcal{PORD}(n,r) =\{ α\in \mathcal{PORD}_{n} :\lvert \text{im}(α) \rvert \leq r\}$ and $\mathcal{IORD}(n,r)=\{ α\in \mathcal{IORD}_{n} :\lvert \text{im}(α)\rvert \leq r\}$. In this paper, we determine some minimal generating sets and ranks of $\mathcal{PORD}(n,r)$ and $\mathcal{IORD}(n,r)$, and moreover, we characterize the maximal subsemigroups of $\mathcal{PORD}(n,r)$ and $\mathcal{IORD}(n,r)$.

On certain semigroups of finite oriented and order-decreasing partial transformations

Abstract

Let be the semigroup consisting of all oriented and order-decreasing partial transformations on the finite chain . Let be the subsemigroup of consisting of all injective transformations of . For , let and . In this paper, we determine some minimal generating sets and ranks of and , and moreover, we characterize the maximal subsemigroups of and .
Paper Structure (4 sections, 27 theorems, 53 equations)

This paper contains 4 sections, 27 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Minimal generating set and rank of $\mathcal{PORD}(n,r)$ and $\mathcal{IORD}(n,r)$
  4. Maximal subsemigroups of $\mathcal{PORD}(n,r)$ and $\mathcal{IORD}(n,r)$

Key Result

Proposition 1

$r_{n}=n-\left \lfloor \frac{n}{3} \right \rfloor$.

Theorems & Definitions (27)

  • Proposition 1
  • Lemma 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Lemma 7
  • Theorem 8
  • Corollary 9
  • Proposition 10
  • ...and 17 more