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Lattice pseudoeffect algebras as double residuated structures

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Abstract

Pseudoeffect algebras are partial algebraic structures which are non-commutative generalizations of effect algebras. The main result of the paper is a characterization of lattice pseudoeffect algebras in terms of so-called pseudo Sasaki algebras. In contrast to pseudoeffect algebras, pseudo Sasaki algebras are total algebras. They are obtained as a generalization of Sasaki algebras, which in turn characterize lattice effect algebras. Moreover, it is shown that lattice pseudoeffect algebras are a special case of double CI-posets, which are algebraic structures with two pairs of residuated operations, and which can be considered as generalizations of residuated posets. For instance, a lattice ordered pseudoeffect algebra, regarded as a double CI-poset, becomes a residuated poset if and only if it is a pseudo MV-algebra. It is also shown that an arbitrary pseudoeffect algebra can be described as a special case of conditional double CI-poset, in which case the two pairs of residuated operations are only partially defined.

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Acknowledgments

The authors are grateful to the referee for careful reading of the manuscript and for valuable comments that helped to improve the paper considerably. The second and third author were supported by Center of Excellence SAS-Quantum Technologies; ERDF OP R&D Projects CE QUTE ITMS 26240120009, and meta-QUTE ITMS 26240120022; the grant VEGA No. 2/0032/09 SAV; the Slovak Research and Development Agency under the contract LPP-0199-07.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA, USA

    David J. Foulis

  2. Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, 814 73, Bratislava, Slovakia

    Sylvia Pulmannová & Elena Vinceková

Authors
  1. David J. Foulis
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  2. Sylvia Pulmannová
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  3. Elena Vinceková
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Correspondence to Sylvia Pulmannová.

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Foulis, D.J., Pulmannová, S. & Vinceková, E. Lattice pseudoeffect algebras as double residuated structures. Soft Comput 15, 2479–2488 (2011). https://doi.org/10.1007/s00500-011-0710-7

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