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Combinatorial results for semigroups of order-preserving mappings

  • Peter M. Higgins (a1)


Consider the finite set Xn = {1,2, …,n} ordered in the standard way. Let Tn denote the full transformation semigroup on Xn, that is, the semigroup of all mappings α: XnXn under composition. We shall call α order-preserving if ij implies iα ≤ jα for i,jXn, and α is decreasing if iα ≤ i for all iXn. This paper investigates combinatorial properties of the semigroup O of all order-preserving mappings on Xn, and of its subsemigroup C which consists of all decreasing and order-preserving mappings.



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[1]Feller, W.. An Introduction to Probability Theory and its Applications, 3rd edn. vol. 1 (Wiley, 1968).
[2]Garba, G. U.. Nilpotents in semigroups of partial one-to-one order-preserving mappings. Semigroup Forum 41 (1991), 115.
[3]Gomes, G. M. S. and Howie, J. M.. On the ranks of certain semigroups of order-preserving transformations. Semigroup Forum 40 (1990), 115.
[4]Harris, B.. A note on the number of idempotents in symmetric semigroups. J. Combin. Theory Ser. A 3 (1967), 12341235.
[5]Harris, B. and Schoenfeld, L.. The number of idempotent elements in symmetric semigroups. J. Combin. Theory Ser. A 3 (1967), 122135.
[6]Higgins, P. M.. The range order of a product of i transformations from a finite full transformation semigroup. Semigroup Forum 37 (1988), 3136.
[7]Higgins, P. M.. Random products in semigroups of mappings. In Lattices, Semigroups and Universal Algebra (editors Almeida, J. et al.), (Plenum Press, 1990), pp. 89100.
[8]Higgins, P. M.. Techniques of Semigroup Theory (Oxford University Press, 1992).
[9]Higgins, P. M. and Williams, E. J.. Random functions on a finite set. Ars Combin. 26A (1988), 93102.
[10]Hilton, P. and Pedersen, J.. Catalan numbers, their generalization, and their uses. Math. Intelligencer 13 (1991), 6475.
[11]Howie, J. M.. Products of idempotents in certain semigroups of transformations. Proc. Roy. Soc. Edinburgh Sect. A 17 (1971), 223236.
[12]Katz, L.. Probability of indecomposability of a random mapping function. Ann. Math. Statist. 26 (1955), 512517.
[13]Kruskal, M. T.. The expected number of components under a random mapping function. Amer. Math. Monthly 61 (1954), 392397.
[14]Pin, J. E.. Varieties of Formal Languages (Plenum Press, 1986).


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