Rank properties of the semigroup reducts of affine near-semirings over Brandt semigroups
type: Item
instance of: scholarly article
title: Rank properties of the semigroup reducts of affine near-semirings over Brandt semigroups
zbMATH Open document ID: 1373.16075
author: Yong-Cai Geng
author: Sumit K. Garg
published in: Semigroup Forum
publication date: 2017-02-24T00:00:00Z
review text: There are several different ways to define ranks of semigroups based on the idea of independent subsets and generating subsets. \textit{J. M. Howie} and \textit{M. I. Marques Ribeiro} considered five such ranks in two papers [Commun. Algebra 27, No. 11, 5333--5347 (1999; Zbl 0940.20056); Southeast Asian Bull. Math. 24, No. 2, 231--237 (2000; Zbl 0967.20030)] denoted as \(r_{i}\), \(i=1,2,3,4,5\). In general, for any finite semigroup \(\Gamma\), \(r_{1}(\Gamma)\leq r_{2}(\Gamma)\leq r_{3}(\Gamma)\leq r_{4}(\Gamma)\leq r_{5}(\Gamma)\), and so they are called by small rank, lower rank, intermediate rank, upper rank and large rank of \(\Gamma\), respectively. For some semigroups, these five ranks are equal. Yet, for some other semigroups, these may be all distinct. In this article under review, the rank properties of the semigroup reducts of the affine near-semiring \(A^{+}(B_{n})\) over the Brandt semigroup \(B_{n}\) are investigated. Either the ranks are explicitly computed, or some bounds for the ranks are given.
reviewed by: Wen-Fong Ke
Mathematics Subject Classification ID: 16Y99
Mathematics Subject Classification ID: 16Y60
Mathematics Subject Classification ID: 16Y30
Mathematics Subject Classification ID: 20M18
zbMATH DE Number: 6688587
zbMATH Keywords: rank properties
zbMATH Keywords: near-semiring
zbMATH Keywords: Brandt semigroup
zbMATH Keywords: affine maps
MaRDI profile type: Publication
full work available at URL: s00233-016-9826-5
OpenAlex ID: W2519298528
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DOI: 10.1007/S00233-016-9826-5
description: scientific article; zbMATH DE number 6688587