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.
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References
Emilia Giraldes, Semigroups of high rank. II. ‘Doubly noble semigroups’, Proc. Edinburgh Math. Soc. 28 (1985) 409–417.
Emilia Giraldes and John M. Howie, ‘Semigroups of high rank’, Proc. Edinburgh Math. Soc. 28 (1985) 13–34.
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© 1987 D. Reidel Publishing Company
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Howie, J.M. (1987). Rank Properties in Semigroups of Mappings. In: Goberstein, S.M., Higgins, P.M. (eds) Semigroups and Their Applications. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-3839-7_8
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DOI: https://doi.org/10.1007/978-94-009-3839-7_8
Publisher Name: Springer, Dordrecht
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