Geometry of the biordered set of idempotent endomorphisms.
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.
scientific article; zbMATH DE number 2067253