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Algebras in the Positive Cone of po-Groups

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Abstract

We systemize a number of algebras that are especially known in the field of quantum structures and that in particular arise from the positive cones of partially ordered groups. Generalized effect algebras, generalized difference posets, cone algebras, commutative BCK-algebras with the relative cancellation property, and positive minimal clans are included in the text.

All these structures are conveniently characterizable as special cases of generalized pseudoeffect algebras, which we introduced in a previous paper. We establish the exact relations between all mentioned structures, thereby adding new structures whenever necessary to make the scheme of order complete.

Generalized pseudoeffect algebras were under certain conditions proved to be representable by means of a po-group. From this fact, we will easily establish representation theorems for all of the structures included in discussion.

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Authors and Affiliations

  1. Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, SK-814 73, Bratislava, Slovakia

    Anatolij Dvurečenskij & Thomas Vetterlein

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  1. Anatolij Dvurečenskij
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  2. Thomas Vetterlein
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Dvurečenskij, A., Vetterlein, T. Algebras in the Positive Cone of po-Groups. Order 19, 127–146 (2002). https://doi.org/10.1023/A:1016551707476

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