Complementation in the face lattice of a proper cone (Q1076701)

Let K be a proper cone in a finite dimensional real space, and let \({\mathfrak F}(K)\) be the lattice of faces. By a result of \textit{G. P. Barker} [Linear Algebra Appl. 7, 71-82 (1973; Zbl 0249.15010)] \({\mathfrak F}(K)\) is complemented. The paper illustrates a gap in the proof of Barker and gives a proof that \({\mathfrak F}(K)\) is even section complemented. Furthermore conditions are given for \({\mathfrak F}(K)\) being relatively complemented.