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On the ranks of certain finite semigroups of transformations

  • Gracinda M. S. Gomes (a1) and John M. Howie (a2)

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It is well-known (see [2]) that the finite symmetric group Sn has rank 2. Specifically, it is known that the cyclic permutations

generate Sn,. It easily follows (and has been observed by Vorob'ev [9]) that the full transformation semigroup on n (< ∞) symbols has rank 3, being generated by the two generators of Sn, together with an arbitrarily chosen element of defect 1. (See Clifford and Preston [1], example 1.1.10.) The rank of Singn, the semigroup of all singular self-maps of {1, …, n}, is harder to determine: in Section 2 it is shown to be ½n(n − 1) (for n ≽ 3). The semigroup Singn it is known to be generated by idempotents [4] and so it is possible to define the idempotent rank of Singn as the cardinality of the smallest possible set P of idempotents for which <F> = Singn. This is of course potentially greater than the rank, but in fact the two numbers turn out to be equal.

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[1]Clifford, A. H. and Preston, G. B.. The algebraic theory of seinigroups, vol. 1 (Math. Surveys of the American Math. Soc. 7, 1961).
[2]Coxeter, H. S. M. and Moser, W. O. J.. Generators and relations for discrete groups, 3rd ed. (Springer, 1972).
[3]Gomes, G. M. S. and Howie, J. M.. Nilpotents in finite symmetric inverse semigroups. Proc. Edinburgh Math. Soc.xs (submitted).
[4]Howie, J. M.. The subsemigroup generated by the idempotents of a full transformation semigroup. J. London Math. Soc. 41 (1966), 707716.
[5]Howie, J. M.. An introduction to semigroup theory (Academic Press, 1976).
[6]Howie, J. M.. Idempotent generators in finite full transformation semigroups. Proc. Royal Soc. Edinburgh A 81 (1978), 317323.
[7]Lyndon, R. C. and Schupp, P. E.. Combinatorial group theory (Springer, 1977).
[8]Petrich, M.. Inverse semi groups (Wiley, 1984).
[9]Vorob'ev, N. N.. On symmetric associative systems. Leningrad Gos. Ped. Inst. Uch. Zap. 89 (1953), 393396 (Russian).
[10]Wilson, R. J.. Introduction to graph theory (Longman, 1972).

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