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Endomorphisms of \({\mathfrak B}_n\), \({\mathcal P}{\mathfrak B}_n\), and \({\mathfrak C}_n\). (English) Zbl 1008.20056

Generalizing the concept of the symmetric group \(S_n\) of degree \(n\), the author considers the Brauer semigroup \({\mathfrak B}_n\) (its definition is too long for this review). The author generalizes \({\mathfrak B}_n\) to \({\mathcal P}{\mathfrak B}_n\). Another generalization of \({\mathfrak B}_n\) was given in [C. Xi, Compos. Math. 119, No. 1, 99-109 (1999; Zbl 0939.16006)]; although it considers algebras rather than semigroups, it is easy to define the corresponding semigroup \({\mathfrak C}_n\). The author finds endomorphisms of these semigroups and proves that all automorphisms of \({\mathfrak C}_n\) are inner.

MSC:

20M20 Semigroups of transformations, relations, partitions, etc.
20M15 Mappings of semigroups

Citations:

Zbl 0939.16006
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References:

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[5] DOI: 10.1006/jabr.1997.7132 · Zbl 0887.20033 · doi:10.1006/jabr.1997.7132
[6] DOI: 10.1090/S0002-9939-98-04764-9 · Zbl 0912.20047 · doi:10.1090/S0002-9939-98-04764-9
[7] DOI: 10.1080/00927879808826385 · Zbl 0918.20051 · doi:10.1080/00927879808826385
[8] Changchang Xi, Compositio Math. 119 pp 99– (1999)
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