Consistent diagonalization, representations, and lattices (Q1174631)

Let \(R\) be a complete discrete valuation domain with quotient field \(K\). The authors investigate systems \(F=(F_ 1,F_ 2,F_ 3)\) of full \(R\)- lattices \(F_ 1\subset F_ 2\subset F_ 3\) in a finite dimensional \(K\)-space up to isomorphism. For \(R=\widehat\mathbb{Z}_ p\), this is equivalent to a problem considered by \textit{G. Birkhoff} [Proc. Lond. Math. Soc., II. Ser. 38, 385-401 (1934; Zbl 0010.34304)]. In their main theorem, the authors give an algorithmic criterion for lattice systems \(F\) with a simultaneous basis. They also suggest a generalization to systems \((F_ 1,\dots,F_ n)\) with arbitrary \(n\) or to systems over a more general ring \(R\), and raise the question for an analogous basis criterion. For \(R\) a hereditary order in a simple algebra, and \(n\in\mathbb{N}\), an answer is given by the reviewer [in Commun. Algebra 9, 893-932 (1981; Zbl 0468.16010), p. 910, Prop. 5.1.].