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The monoid of ordered partitions of a natural number. (English) Zbl 0852.20050

An operation is introduced making \(M\), the set of all sequences of nonnegative integers into a monoid shown to be isomorphic to the monoid of all partial, finite-to-one, order-preserving transformations \(f\) of the positive integers \(\mathbb{N}\), where \(\text{dom }f\) is an initial segment of \(\mathbb{N}\).
A presentation is found for a submonoid \(S_n\) of \(M\) the elements of which are ordered partitions of a natural number \(n\), the posets of principal one-sided ideals of \(S_n\) are described, that for left ideals being the more complex: an algorithm is constructed for the Hasse diagram of the poset of left principal ideals of \(S_n\).

MSC:

20M20 Semigroups of transformations, relations, partitions, etc.
06A07 Combinatorics of partially ordered sets
20M12 Ideal theory for semigroups
11P81 Elementary theory of partitions
20M05 Free semigroups, generators and relations, word problems
05A17 Combinatorial aspects of partitions of integers
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References:

[1] Aîzenŝtat, A. Ja.,Generating relations of an endomorphism semigroup of a finite linearly ordered set, Sibir. Mat. Z.3 (1962), 161–169 (Russian).
[2] Gomes, G. M. S. and Howie, J. M.,On the ranks of certain semigroups of order-preserving transformations, Semigroup Forum41 (1991), 1–15.
[3] Higgins, P.M.,Combinatorial results for semigroups of order-preserving mappings, Math. Proc. Camb. Phil. Soc.113 (1993) 281–296. · Zbl 0781.20036 · doi:10.1017/S0305004100075964
[4] Higgins, P.M., ”Techniques of Semigroup Theory”, Oxford University Press, 1992.
[5] Higgins, P.M.,Divisors of semigroups of order-preserving mappings of a finite chain, Int. J. Algebra and Comp. (to appear). · Zbl 0842.20054
[6] Howie, J.M.,Products of idempotents in certain semigroups of transformations, Proc. Edinburgh Math. Soc.17 (1971), 223–236. · Zbl 0226.20072 · doi:10.1017/S0013091500026936
[7] Howie, J.M., ”An introduction to semigroup theory”, Academic Press, 1976. · Zbl 0355.20056
[8] Howie, J.M.,Semigroups and Combinatorics, in proc. Monash Conference on Semigroup Theory in Honour of G.B. Preston. World Scientific, 1991. · Zbl 1038.20514
[9] Hwang, F.K. and Mallows, C.L.,Enumerating nested and consecutive partitions, J. Comb. Theory, Series A,70 (1995), 323–333. · Zbl 0819.05005 · doi:10.1016/0097-3165(95)90097-7
[10] Lavers, T.G.,Fibonacci numbers, ordered partitions, and transformations of a finite set, Australasian J. of Comb.10 (1994), 147–151. · Zbl 0815.05005
[11] Solomon, A.I.,Catalan Monoids, Monoids of Local Endomorphisms, and their Presentations, Semigroup Forum (to appear). · Zbl 0862.20049
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