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Rank Properties in Semigroups of Mappings

In: Semigroups and Their Applications

Author

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  • John M. Howie

    (University of St Andrews, Mathematical Institute)

Abstract

The rank of a finite semigroup S is defined as r(S) = min{|A|: ‹A› = S}. If S is generated by its set E of idempotents or by its set N of nilpotents, then the idempotent rank ir(S) and the nilpotent rank nr(S) are given by ir(S) = min{|A|:A ⊆ E and ‹A› = S} and nr(S) = min{|A|:A ⊆ n and ‹A› = S} respectively; these are potentially different from r(S). If Singn is the semigroup of all singular self-maps of {1, …, n} then r(Singn) = ir(Singn) = 1/2n(n−1). If SPn is the inverse semigroup of all proper subpermutations of {1, …, n} then r(SPn) = nr(SPn) = n + 1.

Suggested Citation

  • John M. Howie, 1987. "Rank Properties in Semigroups of Mappings," Springer Books, in: Simon M. Goberstein & Peter M. Higgins (ed.), Semigroups and Their Applications, pages 57-60, Springer.
    • Handle: RePEc:spr:sprchp:978-94-009-3839-7_8
    DOI: 10.1007/978-94-009-3839-7_8
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