On weakly graded posets of order-preserving maps under the natural partial order
Commun. Korean Math. Soc.
Published online November 12, 2019
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 : 20M20, 06A06
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