IJPAM: Volume 117, No. 2 (2017)

Title

GENERATING SETS OF SEMIGROUPS OF PARTIAL
TRANSFORMATIONS PRESERVING A ZIG-ZAG ORDER ON $\mathbb{N}$

Authors

Ilinka Dimitrova$^1$, Jörg Koppitz$^2$, Laddawan Lohapan$^3$
$^1$Department of Mathematics
Faculty of Mathematics and Natural Science
South-West University ``Neofit Rilski"
66, Ivan Mihailov Str., Blagoevgrad, 2700, BULGARIA
$^2$Institute of Mathematics and Informatics
Bulgarian Academy of Sciences
8, Acad. G. Bonchev Str., Sofia, 1113, BULGARIA
$^3$Department of Mathematics, Faculty of Science
Khon Kaen University
123 Moo 16 Mittapap Rd., Khon Kaen, 40002, THAILAND

Abstract

This paper deals with the monoid $PF_{\mathbb{N}}$ of all partial transformations on $\mathbb{N}$ preserving a zig-zag order on $\mathbb{N}$. We determine the relative rank of $PF_{\mathbb{N}}$ modulo a set containing all idempotents and all surjections in $PF_{\mathbb{N}}$. Moreover, we show that all transformations in $PF_{\mathbb{N}}$ with finite rank can be generated by the idempotents with finite rank and the full transformation $\gamma_{0}$ with infinite rank, where $\gamma_{0}$ maps each natural number $n$ to $n+2$.

History

Received: 2017-06-14
Revised: 2017-11-15
Published: December 23, 2017

AMS Classification, Key Words

AMS Subject Classification: 20M20
Key Words and Phrases: transformation semigroup, zig-zag order, partial transformations, relative rank, generators

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Bibliography

1
J. D. Currie, T. I. Visentin, The number of order-preserving maps of fences and crowns, Order 8(2) (1991), 133-142.

2
I. Dimitrova, J. Koppitz, On the semigroup of all partial fence-preserving injections on a finite set,Journal of algebra and its applications, 16(11) (2017), 1750223.

3
V. H. Fernades, J. Koppitz, T. Musunthia, On the semigroup of allorder-preserving transformations on a finite set, (2017), preprint.

4
G. M. S Gomes, J. M. Howie, On the ranks of certain semigroups oforder-preserving transformations, Semigroup Forum, 45 (1992), 272-282.

5
P. M. Higgens, J. D. Mitchell, N. Ruškuc, Rank properties ofendomorphisms of infinite partially ordered sets, Bulletin of the LondonMathematical Society, 38 (2006), 177-191.

6
J. M. Howie, N. Ruškuc and P. M. Higgins, On relative ranks of full transformation semigroups,Comm. Algebra, 26 (1998), 733-748.

7
J. Koppitz, L. Lohapan, Regular semigroups of partial transformationspreserving a fence $\mathbb{N}$, Novi Sad Journal of Mathematics, (2016), submitted.

8
N. Ruškuc, On the rank of completely 0-simple semigroups,Math. Proc. Cambridge Philos. Soc., 116 (1994), 325-338.

9
A. Rutkowski, The formula for the number of order-preserving self-mappings of a fence, Order 9(2) (1992), 127-137.

10
R. Tanyawong, R. Srithus, R. Chinram, Regular subsemigroups of the semigroupsof transformations preserving a fence, Asian-European Journal of Mathematics, 9(1) (2016), 1650003.

How to Cite?

DOI: 10.12732/ijpam.v117i2.4 How to cite this paper?

Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2017
Volume: 117
Issue: 2
Pages: 279 - 289


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