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Products of idempotents in finite full transformation semigroups: some improved bounds

  • John M. Howie (a1)

Synopsis

Let E be the set of idempotents in Sn, the semigroup of all singular selfmaps of {1,…, n}. For each α in Sn, there is a unique (κ(α)≧1 such that αψEκ,(α). It is known that κ(α)≦ n + cycl α -fix α, where cyclα is the number of cyclic orbits of a and fix α is the number of fixed points. Equality holds only in the case where a is of rank n – 1. An improved upper bound is obtained for κ(α), applying to elements of arbitrary rank. A lower bound is obtained also.

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1Howie, J. M.. The subsemigroup generated by the idempotents of a full transformation semigroup. J. London Math. Soc. 41 (1966), 707716.
2Howie, J. M.. An introduction to semigroup theory (London: Academic Press, 1976).
3Howie, J. M.. Products of idempotents in finite full transformation semigroups. Proc. Roy. Soc. Edinburgh Sect. A 86 (1980), 243254.
4Iwahori, Nobuko. A length formula in a semigroup of mappings. J. Fac. Sci. Univ. Tokyo Sect. 1A Math. 24 (1977), 255260.

Products of idempotents in finite full transformation semigroups: some improved bounds

  • John M. Howie (a1)

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