Abstract
We characterize effect algebras and pseudo effect algebras that can be represented as an interval of the lexicographic product of an antilattice unital po-group (H, u) with a directed po-group G, both with the Riesz Decomposition Property (RDP). We show that a crucial condition is the existence of a lexicographic normal ideal. Finally, we present a categorical equivalence of the category of (H, u)-lexicographic pseudo effect algebras having RDP with the category of directed po-groups with RDP. In addition, a weaker form of the (H, u)-lexicographic pseudo effect algebra is studied.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Birkhoff G (1967) Lattice theory. Am Math Soc Coll Publ, vol 25, Providence, Rhode Island
Birkhoff G, von Neumann J (1936) The logic of quantum mechanics. Ann Math 37:823–843
Darnel MR (1995) Theory of lattice-ordered groups. Marcel Dekker Inc, New York
Di Nola A, Grigolia R (2015) Gödel spaces and perfect MV-algebras. J Appl Logic 13:270–284
Di Nola A, Lettieri A (1994) Perfect MV-algebras are categorically equivalent to abelian \(\ell \)-groups. Stud Logica 53:417–432
Di Nola A, Lettieri A (1996) Coproduct MV-algebras, nonstandard reals and Riesz spaces. J Algebra 185:605–620
Diaconescu D, Flaminio T, Leuştean I (2014) Lexicographic MV-algebras and lexicographic states. Fuzzy Sets Syst 244:63–85. doi:10.1016/j.fss.2014.02.010
Dvurečenskij A (2002) Pseudo MV-algebras are intervals in \(\ell \)-groups. J Aust Math Soc 72:427–445
Dvurečenskij A (2003) Ideals of pseudo-effect algebras and their applications. Tatra Mt Math Publ 27:45–65
Dvurečenskij A (2007) Perfect effect algebras are categorically equivalent with Abelian interpolation po-groups. J Aust Math Soc 82:183–207
Dvurečenskij A (2015) Lexicographic pseudo MV-algebras. J Appl Logic 13:825–841. doi:10.1016/j.jal.2015.10.001
Dvurečenskij A (2016a) Lexicographic effect algebras. Algebra Univers 75:451–480. doi:10.1007/s00012-016-0374-3
Dvurečenskij A (2016b) Riesz decomposition properties and the lexicographic product of po-groups. Soft Comput 20:2103–2117. doi:10.1007/s00500-015-1903-2
Dvurečenskij A, Vetterlein T (2001) Pseudoeffect algebras. I. Basic properties. Int J Theor Phys 40:685–701
Dvurečenskij A, Vetterlein T (2001) Pseudoeffect algebras. II. Group representation. Int J Theor Phys 40:703–726
Foulis DJ, Bennett MK (1994) Effect algebras and unsharp quantum logics. Found Phys 24:1331–1352
Fuchs L (1963) Partially ordered algebraic systems. Pergamon Press, Oxford
Georgescu G (2001) Iorgulescu A. Pseudo-MV algebras. Multi Valued Logic 6:95–135
Glass AMW (1999) Partially ordered groups. World Scientific, Singapore
Goodearl KR (1986) Partially Ordered Abelian Groups with Interpolation. Math. Surveys Monographs No. 20, Am. Math. Soc., Providence, Rhode Island
Hájek P (2003) Observations on non-commutative fuzzy logic. Soft Comput 8:38–43
Luxemburg WAJ, Zaanen AC (1971) Riesz spaces I. North-Holland, Amsterdam
Mac Lane S (1971) Categories for the working mathematician. Springer, New York
Pulmannová S (2002) Compatibility and decomposition ofeffects. J Math Phys 43:2817–2830
Rachůnek J (2002) A non-commutative generalization of MV-algebras. Czechoslov Math J 52:255–273
Acknowledgments
This study was funded by the Grant VEGA No. 2/0069/16 SAV and GAČR 15-15286S.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflicts of interest
Author declares that he has no conflict of interest.
Human participants or animals
This article does not contain any studies with human participants or animals performed by the author.
Additional information
Communicated by A. Di Nola.
Rights and permissions
About this article
Cite this article
Dvurečenskij, A. Lexicographic pseudo effect algebras. Soft Comput 21, 4981–4994 (2017). https://doi.org/10.1007/s00500-016-2228-5
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00500-016-2228-5