Skip to content
BY-NC-ND 3.0 license Open Access Published by De Gruyter November 15, 2014

States on bounded commutative residuated lattices

  • Michiro Kondo EMAIL logo
From the journal Mathematica Slovaca

Abstract

We define states on bounded commutative residuated lattices and consider their property. We show that, for a bounded commutative residuated lattice X, (1)If s is a state, then X/ker(s) is an MV-algebra.(2)If s is a state-morphism, then X/ker(s) is a linearly ordered locally finite MV-algebra.

Moreover we show that for a state s on X, the following statements are equivalent: (i)s is a state-morphism on X.(ii)ker(s) is a maximal filter of X.(iii)s is extremal on X.

[1] CIGNOLI, R.— D’OTTAVIANO, I.— MUNDICI, D.: Algebraic Foundations of Manyvalued Reasoning, Kluwer Acad. Publ., Dordrecht, 2000. http://dx.doi.org/10.1007/978-94-015-9480-610.1007/978-94-015-9480-6Search in Google Scholar

[2] DVUREČENSKIJ, A.: States on pseudo MV-algebras, Studia Logica 68 (2001), 301–327. http://dx.doi.org/10.1023/A:101249062045010.1023/A:1012490620450Search in Google Scholar

[3] DI NOLA, A.— GEROGESCU, G.— IORGULESCU, A.: Pseudo-BL algebras I; II, Mult.-Valued Logic 8 (2002), 673–714; 717–750. Search in Google Scholar

[4] DVUREČENSKIJ, A.— RACHŮNEK, J.: Probabilistic averaging in bounded commutative residuated ℓ-monoids, Discrete Math. 306 (2006), 1317–1326. http://dx.doi.org/10.1016/j.disc.2005.12.02410.1016/j.disc.2005.12.024Search in Google Scholar

[5] DVUREČENSKIJ, A.— RACHŮNEK, J.: Probabilistic averaging in bounded Rℓ-monoids, Semigroup Forum 72 (2006), 190–206. 10.1007/s00233-005-0545-6Search in Google Scholar

[6] DVUREČENSKIJ, A.— RACHŮNEK, J.: On Riečan and Bosbach states for bounded non-commutative Rℓ-monoids, Math. Slovaca 56 (2006), 487–500. Search in Google Scholar

[7] GEORGESCU, G.: Bosbach states on fuzzy structures, Soft Comput. 8 (2004), 217–230. http://dx.doi.org/10.1007/s00500-003-0266-210.1007/s00500-003-0266-2Search in Google Scholar

[8] HÁJEK, P.: Metamathematics of Fuzzy Logic, Kluwer Acad. Publ., Dordrecht, 1998. http://dx.doi.org/10.1007/978-94-011-5300-310.1007/978-94-011-5300-3Search in Google Scholar

[9] KÔPKA, F.— CHOVANEC, F.: D-Posets, Math. Slovaca 44 (1994), 21–34. Search in Google Scholar

[10] RACHŮNEK, J.— SLEZÁK, J.: Negation in bounded commutative DRℓ-monoids, Czechoslovak Math. J. 56 (2006), 755–763. http://dx.doi.org/10.1007/s10587-006-0053-110.1007/s10587-006-0053-1Search in Google Scholar

[11] TURUNEN, E.— SEESA, S.: Local BL-algebras, Mult.-Valued Logic 6 (2001), 229–249. Search in Google Scholar

Published Online: 2014-11-15
Published in Print: 2014-10-1

© 2014 Mathematical Institute, Slovak Academy of Sciences

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Downloaded on 29.3.2024 from https://www.degruyter.com/document/doi/10.2478/s12175-014-0261-3/html
Scroll to top button