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A combinatorial property of finite full transformation semigroups

  • John M. Howie (a1), Edmund F. Robertson (a1) and Boris M. Schein (a2)

Synopsis

Let E be the set of idempotents in the semigroup Singn of singular self-maps of N = {1, …, n}. Let α ∊ Singn. Then α ∊ E2 if and only if for every x in im α the set −1 either contains x or contains an element of (im α)′.

Write rank α for |im α| and fix α for |{xN: xa = x}|. Define (x, , 2) to be an admissible α-triple if x ∊ (im α)′, xα3xα2. Let comp α (the complexity of α) be the maximum number of disjoint admissible α-triples. Then α ∊ E3 if and only if

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References

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1Harris, Bernard. A note on the number of idempotents in symmetric semigroups. American Math. Monthly 74 (1967), 12341235.
2Harris, Bernard and Schoenfeld, Lowell. The number of idempotent elements in symmetric semigroups. J. Combin. Theory 3 (1967), 122135.
3Howie, J. M.. The subsemigroup generated by the idempotents of a full transformation semigroup. J. London Math. Soc. 41 (1966), 707716.
4Howie, J. M.. Products of idempotents in finite full transformation semigroups. Proc. Roy. Soc. Edinburgh Sect. A 86 (1980), 243254.
5Howie, J. M.. Some subsemigroups of infinite full transformation semigroups. Proc. Roy. Soc. Edinburgh Sect. A 88 (1981), 159167.
6Howie, J. M.. Products of idempotents in finite full transformation semigroups: some improved bounds. Proc. Roy. Soc. Edinburgh Sect. A 98 (1984), 25–35.
7Iwahori, Nobuko. A length formula in a semigroup of mappings. J. Fac. Sci. Univ. Tokyo Sect 1A Math. 24 (1977), 255260.
8Tainiter, M.. A characterization of idempotents in semigroups. J. Combin. Theory 5 (1968), 370373.

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