Idempotents in the universal compactification of a cone (Q1586828)

Let \(S\) be a cone in a real finite-dimensional vector space \(V\). A \textit{face} of \(K\) is a subset \(F\) in \(K\) which is itself a cone and such that if \(x+y\in F\), where \(x\in K\) and \(y\in K\), then \(x\in F\) and \(y\in F\). The intersection of an arbitrary family of faces of \(K\) is again a face of \(K\). Given faces \(A\) and \(B\) of \(K\), \(A\lor B\) denotes the smallest face of \(K\) containing both \(A\) and \(B\). The collection of faces of \(K\) is denoted by \(\mathcal F(K)\). \((\mathcal F(K), \lor, \cap)\) is a complete, complemented, point (atomic) lattice in which no chain exceeds \(1+\text{dim} K\) in length [see \textit{G. P. Barker}, Linear Algebra Appl. 7, 71-82 (1973; Zbl 0249.15010)]. Let \(V^*\) be the dual space of all linear functionals on \(V\). The dual of a proper cone \(K\) is the proper cone \(K^*\subseteq V^*\) defined by \[ K^*=\{ f\in V^*\mid f(x)\geq 0 \text{ for all } x\in K\}. \] \textit{M. Friedberg} [Math. Z. 176, 53-61 (1981; Zbl 0446.22004)] established an isomorphism between the idempotents of the universal (or Bohr) compactification of a real finite-dimensional cone \(K\) and the faces of the cone dual to \(K\). The author utilizes this isomorphism to investigate the generators of the semigroup of idempotents. The main result of the paper is the following \textbf{Theorem.} Let \(U\) be the universal compactification of the cone \(K\) and let \(E\) be the semigroup of idempotents of \(U\). If \(K^*\) is polyhedral or \(\mathcal F(K^*)\) is semimodular, then \(E\) is generated by the maximal elements. Further, the representation is unique if and only if \(K^*\) is simplicial.