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 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 Full-Text :