Previous |  Up |  Next

Article

Keywords:
pseudo $MV$-algebra; $DR\ell $-monoid; generalized pseudo effect algebra
Summary:
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.
References:
[1] M. Anderson and T. Feil: Lattice-Ordered Groups (An Introduction). D. Reidel, Dordrecht, 1988. MR 0937703
[2] P. Bahls, J. Cole, N. Galatos, P. Jipsen and C. Tsinakis: Cancellative residuated lattices. Algebra Univers. 50 (2003), 83–106. MR 2026830
[3] A. Bigard, K. Keimel and S. Wolfenstein: Groupes et Anneaux Réticulés. Springer, Berlin, 1977. MR 0552653
[4] R. Cignoli, I. M. L. D’Ottaviano and D. Mundici: Algebraic Foundations of Many-Valued Reasoning. Kluwer Acad. Publ., Dordrecht, 2000. MR 1786097
[5] A. Dvurečenskij: Pseudo MV-algebras are intervals in $\ell $-groups. J. Austral. Math. Soc. (Ser. A) 72 (2002), 427–445. DOI 10.1017/S1446788700036806 | MR 1902211
[6] A. Dvurečenskij and S. Pulmannová: New Trends in Quantum Structures. Kluwer Acad. Publ., Dordrecht, 2000. MR 1861369
[7] A. Dvurečenskij and J. Rachůnek: Probabilistic averaging in bounded $R\ell $-monoids. Semigroup Forum 72 (2006), 191–206. DOI 10.1007/s00233-005-0545-6 | MR 2216089
[8] A. Dvurečenskij and T. Vetterlein: Pseudo-effect algebras I. Basic properties. Internat. J. Theor. Phys. 40 (2001), 685–701. DOI 10.1023/A:1004192715509 | MR 1865061
[9] A. Dvurečenskij and T. Vetterlein: Pseudo-effect algebras II. Group representations. Internat. J. Theor. Phys. 40 (2001), 703–726. DOI 10.1023/A:1004144832348 | MR 1865061
[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 1865061
[11] N. Galatos and C. Tsinakis: Generalized MV-algebras. J. Algebra 283 (2005), 254–291. DOI 10.1016/j.jalgebra.2004.07.002 | MR 2102083
[12] G. Georgescu and A. Iorgulescu: Pseudo-MV algebras. Mult.-Valued Log. 6 (2001), 95–135. MR 1817439
[13] G. Georgescu, L. Leuştean and V. Preoteasa: Pseudo-hoops. J. Mult.-Val. Log. Soft Comput. 11 (2005), 153–184. MR 2162590
[14] A. M. W. Glass: Partially Ordered Groups. World Scientific, Singapore, 1999. MR 1791008 | Zbl 0933.06010
[15] P. Hájek: Observations on non-commutative fuzzy logic. Soft Comput. 8 (2003), 38–43. DOI 10.1007/s00500-002-0246-y
[16] A. Iorgulescu: Classes of pseudo-BCK(pP) lattices. Preprint. MR 2648142
[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 2083033
[18] J. Kühr: Ideals of noncommutative $DR\ell $-monoids. Czech. Math. J. 55 (2005), 97–111. DOI 10.1007/s10587-005-0006-0 | MR 2121658
[19] J. Kühr: Finite-valued dually residuated lattice-ordered monoids. Math. Slovaca 56 (2006), 397–408. MR 2267761
[20] J. Kühr: On a generalization of pseudo MV-algebras. J. Mult.-Val. Log. Soft Comput 12 (2006), 373–389. MR 2288689
[21] T. Kovář: General Theory of Dually Residuated Lattice Ordered Monoids. Ph.D. thesis, Palacký Univ., Olomouc, 1996.
[22] J. Martinez: Archimedean lattices. Algebra Univers. 3 (1973), 247–260. MR 0349503 | Zbl 0317.06004
[23] D. Mundici: Interpretation of AF C$^*$-algebras in Łukasiewicz sentential calculus. J. Funct. Anal. 65 (1986), 15–63. DOI 10.1016/0022-1236(86)90015-7 | MR 0819173 | Zbl 0597.46059
[24] J. Rachůnek: A non-commutative generalization of MV-algebras. Czech. Math. J. 52 (2002), 255–273. DOI 10.1023/A:1021766309509
[25] J. Rachůnek: Prime spectra of non-commutative generalizations of MV-algebras. Algebra Univers. 48 (2002), 151–169. DOI 10.1007/PL00012447 | MR 1929902
[26] K. L. N. Swamy: Dually residuated lattice ordered semigroups. Math. Ann. 159 (1965), 105–114. DOI 10.1007/BF01360284 | MR 0183797 | Zbl 0138.02104
Partner of
EuDML logo