Abstract
We study conditions when a certain type of the Riesz Decomposition Property (RDP for short) holds in the lexicographic product of two po-groups. Defining two important properties of po-groups, we extend known situations showing that the lexicographic product satisfies RDP or even \({{\rm RDP}_1}\), a stronger type of RDP. We recall that a very strong type of RDP, \({{\rm RDP}_2}\), entails that the group is lattice ordered. RDP's of the lexicographic products are important for the study of lexicographic pseudo effect algebras, or perfect types of pseudo MV-algebras and 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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References
Cignoli, R., D'Ottaviano, I.M.L., Mundici, D.: Algebraic Foundations of Many-valued Reasoning. Kluwer Academic Publ., Dordrecht, (2000)
Darnel, M.R.: Theory of Lattice-Ordered Groups. Marcel Dekker, New York, (1995)
Diaconescu D., Flaminio T., Leuştean I.: Lexicographic MV-algebras and lexicographic states. Fuzzy Sets and Systems 244, 63–85 (2014)
Di Nola A., Grigolia R.: Gödel spaces and perfect MV-algebras. J. Appl. Logic. 13, 270–284 (2015)
Di Nola A., Lettieri A.: Perfect MV-algebras are categorically equivalent to abelian \({\ell}\)-groups. Studia Logica 53, 417–432 (1994)
Di Nola A., Lettieri A.: Coproduct MV-algebras, nonstandard reals and Riesz spaces. J. Algebra 185, 605–620 (1996)
Dvurečenskij A.: Pseudo MV-algebras are intervals \({\ell}\)-groups. J. Austral. Math. Soc. 72, 427–445 (2002)
Dvurečenskij A.: Perfect effect algebras are categorically equivalent with Abelian interpolation po-groups. J. Austral. Math. Soc. 82, 183–207 (2007)
Dvurečenskij A.: Lexicographic pseudo MV-algebras. J. Appl. Logic 13, 825–841 (2015)
Dvurečenskij A.: Riesz decomposition properties and lexicographic product of po-groups. Soft Computing. 20, 2103–2117 (2016)
Dvurečenskij A.: Lexicographic effect algebras. Algebra Universalis 75, 451–480 (2016)
Dvurečenskij A., Kolařík M.: Lexicographic product vs \({\mathbb{Q}}\)-perfect and \({\mathbb{H}}\)-perfect pseudo effect algebras. Soft Computing 17, 1041–1053 (2014)
Dvurečenskij A., Vetterlein T.: Pseudoeffect algebras. I. Basic properties. Inter. J. Theor. Phys. 40, 685–701 (2001)
Dvurečenskij A., Vetterlein T.: Pseudoeffect algebras. II. Group representation. Inter. J. Theor. Phys. 40, 703–726 (2001)
Foulis D.J., Bennett M.K.: Effect algebras and unsharp quantum logics. Found. Phys. 24, 1331–1352 (1994)
Fuchs, L.: Partially Ordered Algebraic Systems. Pergamon Press Oxford, New York (1963)
Fuchs L.: Riesz groups. Annali della Scuola Norm. Sup. Pisa 19, 1–34 (1965)
Georgescu G., Iorgulescu A.: Pseudo-MV algebras. Multi Valued Logic. 6, 95–135 (2001)
Glass, A.M.W.: Partially Ordered Groups. World Scientific. Singapore (1999)
Goodearl, K.R.: Partially Ordered Abelian Groups with Interpolation. Math. Surveys and Monographs No. 20, Amer. Math. Soc. Providence (1986)
Luxemburg, W.A.J., C. Zaanen, A.: Riesz Spaces I. North-Holland, (1971)
Mundici D.: Interpretations of AF C *-algebras in Łukasiewicz sentential calculus. J. Funct. Analysis 65, 15–63 (1986)
Rachůnek J.: A non-commutative generalization of MV-algebras. Czechoslovak Math. J. 52, 255–273 (2002)
Ravindran, K.: On a structure theory of effect algebras. PhD thesis, Kansas State Univ. (1996)
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Presented by S. Pulmannova.
This work was supported by the grant VEGA No. 2/0069/16 SAV, and GAČR 15-15286S.
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Dvurečenskij, A., Zahiri, O. When the lexicographic product of two po-groups has the Riesz decomposition property. Algebra Univers. 78, 67–91 (2017). https://doi.org/10.1007/s00012-017-0447-y
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DOI: https://doi.org/10.1007/s00012-017-0447-y