Abstract
It has been recently shown [4] that the lattice effect algebras can be treated as a subvariety of the variety of so-called basic algebras. The open problem whether all subdirectly irreducible distributive lattice effect algebras are just subdirectly irreducible MV-chains and the horizontal sum \({\mathcal{H}}\) of two 3-element chains is in the paper transferred into a more tractable one. We prove that modulo distributive lattice effect algebras, the variety generated by MV-algebras and \({\mathcal{H}}\) is definable by three simple identities and the problem now is to check if these identities are satisfied by all distributive lattice effect algebras or not.
Similar content being viewed by others
References
Botur, M., and R. Halaš, ‘Finite commutative basic algebras are MV-algebras’, Multiple Valued Logic and Soft Comp. 14, 1–2 (2008), 69–80.
Botur M. and Halaš R. (2007). Complete commutative basic algebras. Order 24(1): 89–105
Chajda, I., and R. Halaš, ‘Varieties of lattice effect algebras generated by four element members, Acta Sci. Math., (Szeged), to appear.
Chajda, I., R. Halaš, and J. Kühr, ‘Many-valued quantum algebras’, Algebra Universalis, to appear.
Cignoli R.L.O., D’Ottaviano I.M. L. and Mundici D. (2000). Algebraic Foundations of Many-valued Reasoning. Kluwer Acad. Publ., Dordrecht
Dvurečenskij A. and Pulmannová S. (2000). New Trends in Quantum Structures. Kluwer Acad. Publ., Dordrecht/Ister Science, Bratislava
Foulis D. and Bennett M. (1994). Effect algebras and unsharp quantum logics. Found. Phys. 24: 1325–1346
Jenča, G., I. Marinová, and Z. Riečanová, ‘Central elements, blocks and sharp elements of lattice effect algebras’, in Proceedings of the third seminar Fuzzy Sets and Quantum Structures, Vyhne, Slovakia (2002), 28–33.
Jenča G. and Riečanová Z. (1999). On sharp elements in lattice ordered effect algebras. BUSEFAL 80: 24–29
Kôpka F. and Chovanec F. (1994). D-posets. Math. Slovaca 44: 21–34
Riečanová Z. (2000). Generalization of blocks for D-lattices and lattice ordered effect algebras. Internat. Journal of Theoret. Physics 39: 231–237
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Halaš, R. The Variety of Lattice Effect Algebras Generated by MV-algebras and the Horizontal Sum of Two 3-element Chains. Stud Logica 89, 19–35 (2008). https://doi.org/10.1007/s11225-008-9115-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11225-008-9115-8