Observables on lexicographic effect algebras (Q2009226)

For the notion of lexicographic effect algebras, see the paper [Algebra Univers. 75, No. 4, 451--480 (2016; Zbl 1350.03046)] by the first author. In the paper under review, lexicographic effect algebras are considered that are intervals in lexicographic products \(H \overrightarrow\times G\), where \(\langle H, u \rangle\) is a unital po-group and \(G\) is a monotone \(\sigma\)-complete po-group with interpolation. An observable on such an effect algebra \(E\) is a kind of \(\sigma\)-homomorphism from the Borel algebra \(\mathcal{B}(\mathbb R)\) into \(E\). A spectral resolution on \(E\) is a certain system of elements \(x_t \in E\) with \(t \in \mathbb R\), which is, in particular, monotonic, left continuous in a sense and has \(0\) as the infimum and \(1\) as the supremum. Every observable \(x\) provides a spectral resolution where \(x_t :=x ((-\infty,t))\); this resolution posseses the so called finiteness property. The main result states that, conversely, every spectral resolution on \(E\) having this property is induced in this way by a unique observable. Several other properties of spectral resolutions are also discussed; there is also a number of illustrating examples.