Communications of the
Korean Mathematical Society
CKMS

ISSN(Print) 1225-1763 ISSN(Online) 2234-3024

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Commun. Korean Math. Soc. 2020; 35(2): 347-358

Online first article November 12, 2019      Printed April 30, 2020

https://doi.org/10.4134/CKMS.c190054

Copyright © The Korean Mathematical Society.

On weakly graded posets of order-preserving maps under the natural partial order

Phichet Jitjankarn

Walailak University

Abstract

In this paper, we simplify the natural partial ordering $\preccurlyeq$ on the semigroup $\Op([n])$ under composition of all order-preserving maps on $[n]=\{1,\ldots,n\}$, and describe its maximal elements. Also, we show that the poset $(\Op([n]),\preccurlyeq)$ is weakly graded and determine when $(\Op([n]),\preccurlyeq)$ has a structure of $({\bf i+1})$-avoidance.

Keywords: Transformation semigroup, partial order, (3+1)-avoidance

MSC numbers: Primary 20M20, 06A06

Supported by: This research has been financially supported by Walailak
University, Thailand (Grant no. WU61114).

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