Geometry of the biordered set of idempotent endomorphisms.
type: Item
instance of: scholarly article
title: Geometry of the biordered set of idempotent endomorphisms.
zbMATH Open document ID: 1054.20045
author: V. N. Krishnachandran
author: K. S. S. Nambooripad
published in: Southeast Asian Bulletin of Mathematics
publication date: 2004-05-27T00:00:00Z
review text: The geometry of the biordered set \(\mathbb{E}_n\) of idempotent endomorphisms of an \(n\)-dimensional real or complex vector space \(V\) is studied. It is shown that, for \(e\in\mathbb{E}_n\), \(E(L_e)\) and \(E(R_e)\) are maximal affine spaces contained in \(\mathbb{E}_n\), where \(L_e\) (respectively, \(R_e\)) is the \(\mathcal L\)-class (respectively, \(\mathcal R\)-class) of \(e\), and \(E(X)\) is the idempotents in \(X\). Denote by \(E(k)\) the space of idempotent endomorphisms of rank \(k\). It is shown that any maximal rectangular band contained in \(E(k)\) is a point-wise product of two affine spaces whose dimension sum is \(k(n-k)\). It is also shown that \(E(k)\) is a `generalized hyperboloid' in the sense that the space \(E(k)\) has most of the properties of a hyperboloid of one sheet.
reviewed by: Huang Wenxue
Mathematics Subject Classification ID: 20M20
Mathematics Subject Classification ID: 20M17
Mathematics Subject Classification ID: 15A04
zbMATH DE Number: 2067253
zbMATH Keywords: affine spaces
zbMATH Keywords: biordered sets
zbMATH Keywords: hyperboloids
zbMATH Keywords: idempotent endomorphisms
zbMATH Keywords: rectangular bands
zbMATH Keywords: regular semigroups
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2067253