On nilpotent subsemigroups of a finite symmetric inverse semigroup
type: Item
instance of: scholarly article
title: On nilpotent subsemigroups of a finite symmetric inverse semigroup
zbMATH Open document ID: 0837.20075
DOI: 10.1007/BF02362408
author: A. G. Ganyushkin
author: T. V. Kormysheva
published in: Mathematical Notes
publication date: 1996-03-13T00:00:00Z
review text: Let \(N = \{1, 2, \dots, n\}\) and denote by \(JS_n\) the symmetric inverse semigroup of all injective partial selfmaps of \(N\). A partition \(\rho = (N_1, N_2, \dots, N_k)\) into \(k\) nonempty blocks is said to be ordered if the blocks are strictly linearly ordered, that is, if \(N_1 < N_2 < \cdots < N_k\). The ordered \(k\)-tuple \((|N_1|, |N_2|, \dots, |N_k|)\) is said to be the type of the ordered partition \(\rho\) and the type \((n_k, n_{k - 1}, \dots, n_1)\) is said to be opposite to the type \((n_1, n_2, \dots, n_k)\). The authors prove that there is a one-to-one correspondence between the maximal nilpotent subsemigroups of \(JS_n\) of nilpotency level \(k\) and the ordered partitions of \(N\) into \(k\) blocks. They show further that two maximal subsemigroups of \(JS_n\) of nilpotency level \(k \geq 3\) are isomorphic if and only if the types of the corresponding ordered partitions coincide and they are anti- isomorphic if and only if those types are opposite.
Mathematics Subject Classification ID: 20M20
Mathematics Subject Classification ID: 05A18
Mathematics Subject Classification ID: 20M18
zbMATH DE Number: 830723
zbMATH Keywords: symmetric inverse semigroup
zbMATH Keywords: injective partial selfmaps
zbMATH Keywords: blocks
zbMATH Keywords: maximal nilpotent subsemigroups
zbMATH Keywords: ordered partitions
reviewed by: jun. Kenneth D. Magill
MaRDI profile type: Publication
cites work: Q5652898
full work available at URL: https://doi.org/10.1007/bf02362408
OpenAlex ID: W2017653717
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description: scientific article; zbMATH DE number 830723