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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.
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참고문헌 (19)
참고문헌 신청
M. D. Atkinson / 2012 / Counting(3 + 1)-avoiding permutations / European J. Combin 33(1):49~61
T. Brylawski / 1973 / The lattice of integer partitions / Discrete Math 6:201~219
S. Chaopraknoi / 2014 / The natural partial order on some transformation semigroups / Bull. Aust. Math. Soc 89(2):279~292
S. Chaopraknoi / 2015 / The natural partial order on linear semigroups with nullity and co-rank bounded below / Bull. Aust. Math. Soc 91(1):104~115
C. Greene / 1986 / Longest chains in the lattice of integer partitions ordered by majorization / European J. Combin 7(1):1~10
T. Brylawski / 1973 / The lattice of integer partitions / Discrete Math 6:201~219
S. Chaopraknoi / 2014 / The natural partial order on some transformation semigroups / Bull. Aust. Math. Soc 89(2):279~292
S. Chaopraknoi / 2015 / The natural partial order on linear semigroups with nullity and co-rank bounded below / Bull. Aust. Math. Soc 91(1):104~115
C. Greene / 1986 / Longest chains in the lattice of integer partitions ordered by majorization / European J. Combin 7(1):1~10
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