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Compressible groups with general comparability. (English) Zbl 1114.06012
Summary: Compressible groups generalize the order-unit space of self-adjoint operators on Hilbert space, the directed additive group of self-adjoint elements of a unital \(C^*\)-algebra, and interpolation groups with order units. In a compressible group with general comparability, each element \(g\) may be written canonically as a difference \(g = g^+ - g^-\) of elements in the positive cone \(G^+\), and the absolute value \(| g| \) is defined by \(| g| :=g^+ + g^-\). In such a group \(G\), we define and study a “pseudo-meet” \(g\mathbin {\sqcap } h\) and a “pseudo-join” \(g\mathbin {\sqcup } h\). If \(G\) is lattice-ordered, \(g\mathbin {\sqcap } h\) and \(g\mathbin {\sqcup } h\) coincide with the usual meet and join; in the general case, they retain a number of properties of the latter. We also introduce and study a so-called Rickart projection property suggested by an analogous property in Rickart \(C^*\)-algebras.

06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
03G12 Quantum logic
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
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