## Generalizations of pseudo MV-algebras and generalized pseudo effect algebras.(English)Zbl 1174.06330

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 with 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.

### MSC:

 06F05 Ordered semigroups and monoids 06D35 MV-algebras

Pseudo Hoops
Full Text:

### References:

 [1] M. Anderson and T. Feil: Lattice-Ordered Groups (An Introduction). D. Reidel, Dordrecht, 1988. · Zbl 0636.06008 [2] P. Bahls, J. Cole, N. Galatos, P. Jipsen and C. Tsinakis: Cancellative residuated lattices. Algebra Univers. 50 (2003), 83–106. · Zbl 1092.06012 [3] A. Bigard, K. Keimel and S. Wolfenstein: Groupes et Anneaux Réticulés. Springer, Berlin, 1977. · Zbl 0384.06022 [4] R. Cignoli, I. M. L. D’Ottaviano and D. Mundici: Algebraic Foundations of Many-Valued Reasoning. Kluwer Acad. Publ., Dordrecht, 2000. · Zbl 0937.06009 [5] A. Dvurečenskij: Pseudo MV-algebras are intervals in -groups. J. Austral. Math. Soc. (Ser. A) 72 (2002), 427–445. · Zbl 1027.06014 [6] A. Dvurečenskij and S. Pulmannová: New Trends in Quantum Structures. Kluwer Acad. Publ., Dordrecht, 2000. · Zbl 0987.81005 [7] A. Dvurečenskij and J. Rachğnek: Probabilistic averaging in bounded R-monoids. Semi-group Forum 72 (2006), 191–206. · Zbl 1105.06010 [8] A. Dvurečenskij and T. Vetterlein: Pseudo-effect algebras I. Basic properties. Internat. J. Theor. Phys. 40 (2001), 685–701. · Zbl 0994.81008 [9] A. Dvurečenskij and T. Vetterlein: Pseudo-effect algebras II. Group representations. Internat. J. Theor. Phys. 40 (2001), 703–726. · Zbl 0994.81009 [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. · Zbl 1012.03063 [11] N. Galatos and C. Tsinakis: Generalized MV-algebras. J. Algebra 283 (2005), 254–291. · Zbl 1063.06008 [12] G. Georgescu and A. Iorgulescu: Pseudo-MV algebras. Mult.-Valued Log. 6 (2001), 95–135. · Zbl 1014.06008 [13] G. Georgescu, L. Leuştean and V. Preoteasa: Pseudo-hoops. J. Mult.-Val. Log. Soft Comput. 11 (2005), 153–184. · Zbl 1078.06007 [14] A. M. W. Glass: Partially Ordered Groups. World Scientific, Singapore, 1999. [15] P. Hájek: Observations on non-commutative fuzzy logic. Soft Comput. 8 (2003), 38–43. · Zbl 1075.03009 [16] A. Iorgulescu: Classes of pseudo-BCK(pP) lattices. Preprint. [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. · Zbl 1070.06005 [18] J. Kühr: Ideals of noncommutative DR-monoids. Czech. Math. J. 55 (2005), 97–111. · Zbl 1081.06017 [19] J. Kühr: Finite-valued dually residuated lattice-ordered monoids. Math. Slovaca 56 (2006), 397–408. · Zbl 1141.06014 [20] J. Kühr: On a generalization of pseudo MV-algebras. J. Mult.-Val. Log. Soft Comput 12 (2006), 373–389. · Zbl 1131.06005 [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. · Zbl 0272.06013 [23] D. Mundici: Interpretation of AF C*-algebras in Łukasiewicz sentential calculus. J. Funct. Anal. 65 (1986), 15–63. · Zbl 0597.46059 [24] J. Rachunek: A non-commutative generalization of MV-algebras. Czech. Math. J. 52 (2002), 255–273. · Zbl 1012.06012 [25] J. Rachunek: Prime spectra of non-commutative generalizations of MV-algebras. Algebra Univers. 48 (2002), 151–169. · Zbl 1058.06015 [26] K. L. N. Swamy: Dually residuated lattice ordered semigroups. Math. Ann. 159 (1965), 105–114. · Zbl 0135.04203
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