Abstract
The natural bijective correspondence between MV-algebras and pseudorings is generalized to the case of pseudo MV-algebras. We first introduce the concepts of non-commutative pseudorings and strong pseudo De Morgan algebras. Then, we show that they both correspond to pseudo MV-algebras via symmetric difference in a natural bijective way.
References
Baudot R (2000) Non-commutative programming language Noclog In: Symposium LICS. Santa Barbara (Short Presentations), pp 3–9
Chajda I (1996) Pseudosemirings induced by ortholattices. Czechoslov Math. J 46: 405–411
Chajda I, Länger H (2004) Orthorings. Discuss Math Gen Algebra Appl 24: 137–147
Chajda I, Länger H (2004) Ring-like structures corresponding to MV algebras via symmetric difference. Sitz Abt II 213: 33–41
Chajda I, Länger H, Mzczyński M (2004) Ring-like structures corresponding to generalized orthomodular lattices. Math Slovaca 54: 143–150
Dorfer G, Dvurečenskij A, Länger H (1996) Symmetric difference in orthomodular lattices. Math Slovaca 46: 435–444
Dorninger D, Länger H, Mzczyński M (1997) The logic induced by a system of homomorphisms and its various algebraic characterizations. Demonstr Math 30: 215–232
Dorninger D, Länger H, Mzczyński M (1999) On ring-like structures induced by Mackeys probability function. Rep Math Phys 43: 499–515
Dorninger D, Länger H, Mzczyński M (2001) Concepts of measures on ring-like quantum logics. Rep Math Phys 47: 67–176
Dvurečenskij A (2002) Pseudo-MV algebras are intervals in l-groups. J Austral Math Soc 72: 427–445
Dvurečenskij A, Pulmannová S (2000) New trends in quantum structures. Kluwer, Dordrecht
Georgescu G, Iorgulescu A (2001) Pseudo-MV algebras. Multi Valued Logic 6: 95–135
Hajek P (2003) Observations on non-commutative fuzzy logic. Soft Comput 8: 38–43
Länger H (1998) Generalization of the correspondence between Boolean algebras and Boolean rings to orthomodular lattices. Tatra Mt Math Publ 15: 97–105
Rachunek J (2002) A non-commutative generalized of MV-algebras. Czechoslov Math J 52: 255–273
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by NSFC Major Research Program 60496324; NSFC Key Research Program 60736011; NSFC 60603002; Pre-973 Project 2001CCA03000; 863 High-Tech Project 2001AA113130; 973 Project 2001CB312004; CAS Brain and Mind Science Project.
Rights and permissions
About this article
Cite this article
Shang, Y. Ring-like structures corresponding to pseudo MV-algebras. Soft Comput 13, 71–76 (2009). https://doi.org/10.1007/s00500-008-0294-z
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00500-008-0294-z