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Effect algebras are conditionally residuated structures. (English) Zbl 1247.03134

An effect algebra is a partial algebra with a commutative partial operation \(+\) of addition, which was introduced by D. J. Foulis and M. K. Bennett [Found. Phys. 24, No. 10, 1331–1352 (1994; Zbl 1213.06004)]. Since every MV-algebra can be viewed as an effect algebra, which is a commutative residuated structure, there is a problem when effect algebras can also be viewed as partial structures. The authors show that it is possible to deal with them as conditionally residuated structures.
The same is done also for a noncommutative generalization of effect algebras, called pseudoeffect algebras, which were introduced in [A. Dvurečenskij and T. Vetterlein, Int. J. Theor. Phys. 40, No. 3, 685–701 (2001; Zbl 0994.81008)].

MSC:

03G12 Quantum logic
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References:

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