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The transition to unigroups. (English) Zbl 0904.06013
Summary: A unigroup is defined to be a partially ordered Abelian group with a distinguished generative universal order unit. Virtually any structure that has been proposed for the logic, sharp or unsharp, of a physical system can be represented by the order interval in a unigroup. Furthermore, probability states correspond to positive, normalized, real-valued group homomorphisms, and physical symmetries correspond to unigroup automorphisms. We show that the category of unigroups admits arbitrary products and coproducts. A new class of interval effect algebras called Heyting effect algebras (HEAs) is introduced and studied. Among other things, an HEA is both a Heyting algebra and a BZ-lattice in which the sharp elements are precisely the central elements. Certain HEAs arise naturally from partially ordered Abelian groups affiliated with Stone spaces. Using Stone unigroups, we obtain perspicuous representations for certain multivalued logics, including the three-valued logic of conditional events utilized by Goodman, Nguyen, and Walker in their study of logic for expert systems.

MSC:
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
03G12 Quantum logic
03B50 Many-valued logic
68T27 Logic in artificial intelligence
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