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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.

##### MSC:
 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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