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D n -normal semigroups of transformations

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Abstract

Given a subgroup G of the symmetric group S n on n letters, a semigroup S of transformations of X n is G-normal if G S =G, where G S consists of all permutations hS n such that h −1 fhS for all fS. A semigroup S is G-normax if it is a maximal semigroup in the set of all G-normal semigroups.

In 1996, I. Levi showed that the alternating group A n can not serve as the group G S for any semigroup of total transformations of X n . In 2000 and 2001, I. Levi, D.B. McAlister and R.B. McFadden described all A n -normal semigroups of partial transformations of X n . Also, in 1994, I. Levi and R.B. McFadden described all S n -normal semigroups.

In this paper, we show that the dihedral group D n may serve as the group G S for semigroups of transformations of X n . We characterize a large class of D n -normax semigroups and describe certain D n -normal semigroups.

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Correspondence to Inessa Levi.

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Communicated by Thomas E. Hall.

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Catarino, P., Levi, I. D n -normal semigroups of transformations. Semigroup Forum 76, 368–378 (2008). https://doi.org/10.1007/s00233-007-9040-6

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  • DOI: https://doi.org/10.1007/s00233-007-9040-6

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