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Generalizations of pseudo MV-algebras and generalized pseudo effect algebras
Czechoslovak Mathematical Journal, Tome 58 (2008) no. 2, pp. 395-415 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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We deal with unbounded dually residuated lattices that generalize pseudo $MV$-algebras in such a way that every principal order-ideal is a pseudo $MV$-algebra. We describe the connections of these generalized pseudo $MV$-algebras to generalized pseudo effect algebras, which allows us to represent every generalized pseudo $MV$-algebra $A$ by means of the positive cone of a suitable $\ell $-group $G_A$. We prove that the lattice of all (normal) ideals of $A$ and the lattice of all (normal) convex $\ell $-subgroups of $G_A$ are isomorphic. We also introduce the concept of Archimedeanness and show that every Archimedean generalized pseudo $MV$-algebra is commutative.
We deal with unbounded dually residuated lattices that generalize pseudo $MV$-algebras in such a way that every principal order-ideal is a pseudo $MV$-algebra. We describe the connections of these generalized pseudo $MV$-algebras to generalized pseudo effect algebras, which allows us to represent every generalized pseudo $MV$-algebra $A$ by means of the positive cone of a suitable $\ell $-group $G_A$. We prove that the lattice of all (normal) ideals of $A$ and the lattice of all (normal) convex $\ell $-subgroups of $G_A$ are isomorphic. We also introduce the concept of Archimedeanness and show that every Archimedean generalized pseudo $MV$-algebra is commutative.
Classification : 03G25, 06F05
Keywords: pseudo $MV$-algebra; $DR\ell $-monoid; generalized pseudo effect algebra 
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     url = {http://geodesic.mathdoc.fr/item/CMJ_2008_58_2_a6/}
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Kühr, Jan. Generalizations of pseudo MV-algebras and generalized pseudo effect algebras. Czechoslovak Mathematical Journal, Tome 58 (2008) no. 2, pp. 395-415. http://geodesic.mathdoc.fr/item/CMJ_2008_58_2_a6/

[1] M. Anderson and T. Feil: Lattice-Ordered Groups (An Introduction). D. Reidel, Dordrecht, 1988. | MR

[2] P. Bahls, J. Cole, N. Galatos, P. Jipsen and C. Tsinakis: Cancellative residuated lattices. Algebra Univers. 50 (2003), 83–106. | MR

[3] A. Bigard, K. Keimel and S. Wolfenstein: Groupes et Anneaux Réticulés. Springer, Berlin, 1977. | MR

[4] R. Cignoli, I. M. L. D’Ottaviano and D. Mundici: Algebraic Foundations of Many-Valued Reasoning. Kluwer Acad. Publ., Dordrecht, 2000. | MR

[5] A. Dvurečenskij: Pseudo MV-algebras are intervals in $\ell $-groups. J. Austral. Math. Soc. (Ser. A) 72 (2002), 427–445. | DOI | MR

[6] A. Dvurečenskij and S. Pulmannová: New Trends in Quantum Structures. Kluwer Acad. Publ., Dordrecht, 2000. | MR

[7] A. Dvurečenskij and J. Rachůnek: Probabilistic averaging in bounded $R\ell $-monoids. Semigroup Forum 72 (2006), 191–206. | DOI | MR

[8] A. Dvurečenskij and T. Vetterlein: Pseudo-effect algebras I. Basic properties. Internat. J. Theor. Phys. 40 (2001), 685–701. | DOI | MR

[9] A. Dvurečenskij and T. Vetterlein: Pseudo-effect algebras II. Group representations. Internat. J. Theor. Phys. 40 (2001), 703–726. | DOI | MR

[10] A. Dvurečenskij and T. Vetterlein: Generalized pseudo-effect algebras. In: Lectures on Soft Computing and Fuzzy Logic (A. Di Nola, G. Gerla, eds.), Springer, Berlin, 2001, pp. 89–111. | MR

[11] N. Galatos and C. Tsinakis: Generalized MV-algebras. J. Algebra 283 (2005), 254–291. | DOI | MR

[12] G. Georgescu and A. Iorgulescu: Pseudo-MV algebras. Mult.-Valued Log. 6 (2001), 95–135. | MR

[13] G. Georgescu, L. Leuştean and V. Preoteasa: Pseudo-hoops. J. Mult.-Val. Log. Soft Comput. 11 (2005), 153–184. | MR

[14] A. M. W. Glass: Partially Ordered Groups. World Scientific, Singapore, 1999. | MR | Zbl

[15] P. Hájek: Observations on non-commutative fuzzy logic. Soft Comput. 8 (2003), 38–43. | DOI

[16] A. Iorgulescu: Classes of pseudo-BCK(pP) lattices. Preprint. | MR

[17] P. Jipsen and C. Tsinakis: A survey of residuated lattices. In: Ordered Algebraic Structures (J. Martines, ed.), Kluwer Acad. Publ., Dordrecht, 2002, pp. 19–56. | MR

[18] J. Kühr: Ideals of noncommutative $DR\ell $-monoids. Czech. Math. J. 55 (2005), 97–111. | DOI | MR

[19] J. Kühr: Finite-valued dually residuated lattice-ordered monoids. Math. Slovaca 56 (2006), 397–408. | MR

[20] J. Kühr: On a generalization of pseudo MV-algebras. J. Mult.-Val. Log. Soft Comput 12 (2006), 373–389. | MR

[21] T. Kovář: General Theory of Dually Residuated Lattice Ordered Monoids. Ph.D. thesis, Palacký Univ., Olomouc, 1996.

[22] J. Martinez: Archimedean lattices. Algebra Univers. 3 (1973), 247–260. | MR | Zbl

[23] D. Mundici: Interpretation of AF C$^*$-algebras in Łukasiewicz sentential calculus. J. Funct. Anal. 65 (1986), 15–63. | DOI | MR | Zbl

[24] J. Rachůnek: A non-commutative generalization of MV-algebras. Czech. Math. J. 52 (2002), 255–273. | DOI | MR

[25] J. Rachůnek: Prime spectra of non-commutative generalizations of MV-algebras. Algebra Univers. 48 (2002), 151–169. | DOI | MR

[26] K. L. N. Swamy: Dually residuated lattice ordered semigroups. Math. Ann. 159 (1965), 105–114. | DOI | MR | Zbl