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Riesz decomposition properties and the lexicographic product of po-groups

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Abstract

We establish conditions when a certain type of the Riesz decomposition property (RDP) holds in the lexicographic product of two po-groups. It is well known that the resulting product is an \(\ell \)-group if and only if the first one is linearly ordered and the second one is an \(\ell \)-group. This can be equivalently studied as po-groups with a special type of the RDP. In the paper we study three different types of RDPs. RDPs of the lexicographic products are important for the study of pseudo effect algebras where infinitesimal elements play an important role both for algebras as well as for the first-order logic of valid but not provable formulas.

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Acknowledgments

The author is very indebted to an anonymous referee for his/her careful reading and suggestions which helped to improve the readability of the paper.

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Authors and Affiliations

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

    Anatolij Dvurečenskij

  2. Department of Algebra and Geometry, Palacký University, 17. listopadu 12, 771 46, Olomouc, Czech Republic

    Anatolij Dvurečenskij

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  1. Anatolij Dvurečenskij
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Correspondence to Anatolij Dvurečenskij.

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The author declares that there is no conflict of interests regarding the publication of this paper.

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Communicated by A. Di Nola.

This work was supported by the Slovak Research and Development Agency under contract APVV-0178-11, Grant VEGA No. 2/0059/12 SAV, and GAČR 15-15286S.

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Dvurečenskij, A. Riesz decomposition properties and the lexicographic product of po-groups. Soft Comput 20, 2103–2117 (2016). https://doi.org/10.1007/s00500-015-1903-2

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  • DOI: https://doi.org/10.1007/s00500-015-1903-2

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