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Idempotent generators in finite full transformation semigroups*

  • J. M. Howie (a1)

Synopsis

It was proved by Howie in 1966 that , the semigroup of all singular mappings of a finite set X into itself, is generated by its idempotents. Implicit in the method of proof, though not formally stated, is the result that if |X| = n then the n(n – 1) idempotents whose range has cardinal n – 1 form a generating set for. Here it is shown that if n ≧ 3 then a minimal set M of idempotent generators for contains ½n(n–1) members. A formula is given for the number of distinct sets M.

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References

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1Harary, F.Graph theory. (Reading, Mass.: Addison-Wesley, 1969).
2Harary, F. and Palmer, E. M.Graphical enumeration. (New York: Academic Press, 1973).
3Howie, J. M.An introduction to semigroup theory. (London: Academic Press, 1976).
4Howie, J. M.The subsemigroup generated by the idempotents of a full transformation semigroup. J. London Math. Soc. 41 (1966), 707716.
5Moon, John W.Topics on tournaments. (New York: Holt, Rinehart and Winston, 1968).

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