Abstract
A unigroup is defined to be a partially orderedabelian group with a distinguished generative universalorder unit. Virtually any structure that has beenproposed for the logic, sharp or unsharp, of a physical system can be represented by the order intervalin a unigroup. Furthermore, probability statescorrespond to positive, normalized, real-valued grouphomomorphisms, and physical symmetries correspond to unigroup automorphisms. We show that thecategory of unigroups admits arbitrary products andcoproducts. A new class of interval effect algebrascalled Heyting effect algebras (HEAs) is introduced andstudied. Among other things, an HEA is both a Heytingalgebra and a BZ-lattice in which the sharp elements areprecisely the central elements. Certain HEAs arisenaturally from partially ordered abelian groupsaffiliated with Stone spaces. Using Stone unigroups, weobtain perspicuous representations for certainmultivalued logics, including the three-valued logic ofconditional events utilized by Goodman, Nguyen, andWalker in their study of logic for expertsystems.
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Foulis, D.J., Greechie, R.J. & Bennett, M.K. The Transition to Unigroups. International Journal of Theoretical Physics 37, 45–63 (1998). https://doi.org/10.1023/A:1026657004880
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DOI: https://doi.org/10.1023/A:1026657004880