Article
Full entry |
PDF
(0.2 MB)
Feedback
Keywords:
obstinate state filter; prime state filter; Boolean state filter; primary state filter; state filter; residuated lattice; local residuated lattice
obstinate state filter; prime state filter; Boolean state filter; primary state filter; state filter; residuated lattice; local residuated lattice
Summary:
We consider properties of state filters of state residuated lattices and prove that for every state filter $F$ of a state residuated lattice $X$: \begin {itemize} \item [(1)] $F$ is obstinate $\Leftrightarrow $ $L/F \cong \{0,1\}$; \item [(2)] $F$ is primary $\Leftrightarrow $ $L/F$ is a state local residuated lattice; \end {itemize} and that every g-state residuated lattice $X$ is a subdirect product of $\{X/P_{\lambda } \}$, where $P_{\lambda }$ is a prime state filter of $X$. \endgraf Moreover, we show that the quotient MTL-algebra $X/P$ of a state residuated lattice $X$ by a state prime filter $P$ is not always totally ordered, although the quotient MTL-algebra by a prime filter is totally ordered.
References:
[1] Ciungu, L. C.: Bosbach and Riečan states on residuated lattices. J. Appl. Funct. Anal. 3 (2008), 175-188. MR 2369429 | Zbl 1170.03030
[2] Constantinescu, N. M.: On pseudo BL-algebras with internal state. Soft Comput. 16 (2012), 1915-1922. DOI 10.1007/s00500-012-0864-y | Zbl 1291.03116
[3] Constantinescu, N. M.: State filters on fuzzy structures with internal states. Soft Comput. 18 (2014), 1841-1852. DOI 10.1007/s00500-014-1277-x | Zbl 1331.03043
[4] Dvurečenskij, A.: States on pseudo MV-algebras. Stud. Log. 68 (2001), 301-327. DOI 10.1023/A:1012490620450 | MR 1865858 | Zbl 0999.06011
[5] Dvurečenskij, A., Rachůnek, J.: On Riečan and Bosbach states for bounded non-commutative R$\ell$-monoids. Math. Slovaca 56 (2006), 487-500. MR 2293582 | Zbl 1141.06005
[6] Dvurečenskij, A., Rachůnek, J.: Probabilistic averaging in bounded R$\ell$-monoids. Semigroup Forum 72 (2006), 190-206. DOI 10.1007/s00233-005-0545-6 | MR 2216089 | Zbl 1105.06010
[7] Dvurečenskij, A., Rachůnek, J., Šalounová, D.: State operators on generalizations of fuzzy structures. Fuzzy Sets Syst. 187 (2012), 58-76. DOI 10.1016/j.fss.2011.05.023 | MR 2851996 | Zbl 1266.03071
[8] Flaminio, T., Montagna, F.: MV-algebras with internal states and probabilistic fuzzy logics. Int. J. Approx. Reasoning 50 (2009), 138-152. DOI 10.1016/j.ijar.2008.07.006 | MR 2519034 | Zbl 1185.06007
[9] Galatos, N., Jipsen, P., Kowalski, T., Ono, H.: Residuated Lattices: An Algebraic Glimpse at Substructural Logics. Studies in Logic and the Foundations of Mathematics 151. Elsevier, Amsterdam (2007). DOI 10.1016/S0049-237X(07)80005-X | MR 2531579 | Zbl 1171.03001
[10] Georgescu, G.: Bosbach states on fuzzy structures. Soft Comput. 8 (2004), 217-230. DOI 10.1007/s00500-003-0266-2 | Zbl 1081.06012
[11] Hájek, P.: Metamathematics of Fuzzy Logic. Trends in Logic--Studia Logica Library 4. Kluwer, Dordrecht (1998). DOI 10.1007/978-94-011-5300-3 | MR 1900263 | Zbl 0937.03030
[12] Hart, J. B., Rafter, L., Tsinakis, C.: The structure of commutative residuated lattices. Int. J. Algebra Comput. 12 (2002), 509-524. DOI 10.1142/S0218196702001048 | MR 1919685 | Zbl 1011.06006
[13] Haveshki, M., Saeid, A. Borumand, Eslami, E.: Some types of filters in BL algebras. Soft Comput. 10 (2006), 657-664. DOI 10.1007/s00500-005-0534-4 | Zbl 1103.03062
[14] Haveshki, M., Mohamadhasani, M.: Extended filters in bounded commutative R$\ell$-monoids. Soft Comput. 16 (2012), 2165-2173. DOI 10.1007/s00500-012-0884-7 | Zbl 1288.03043
[15] He, P., Xin, X., Yang, Y.: On state residuated lattices. Soft Comput. 19 (2015), 2083-2094. DOI 10.1007/s00500-015-1620-x | Zbl 1364.06003
[16] Kondo, M.: Characterization of extended filters in residuated lattices. Soft Comput. 18 (2014), 427-432. DOI 10.1007/s00500-013-1100-0 | Zbl 1386.03073
[17] Kondo, M.: States on bounded commutative residuated lattices. Math. Slovaca 64 (2014), 1093-1104. DOI 10.2478/s12175-014-0261-3 | MR 3277839 | Zbl 1342.06009
[18] Kondo, M.: Generalized state operators on residuated lattices. Soft Comput. 21 (2017), 6063-6071. DOI 10.1007/s00500-016-2324-6 | Zbl 1384.03119
[19] Kondo, M., Kawaguchi, M. F.: Some properties of generalized state operators on residuated lattices. Proceedings of the 46th IEEE International Symposium on Multiple-Valued Logic IEEE Computer Society, Los Alamitos (2016), 162-166. DOI 10.1109/ISMVL.2016.29 | MR 3570629
[20] Kondo, M., Watari, O., Kawaguchi, M. F., Miyakoshi, M.: A Logic Determined by Commutative Residuated Lattices. New Dimensions in Fuzzy Logic and Related Technologies. Proceedings of the 5th EUSFLAT Conference, Volume 2 (2007), Universitas Ostraviensis, Ostrava (2007), 45-48.
[21] Kôpka, F., Chovanec, F.: D-posets. Math. Slovaca 44 (1994), 21-34. MR 1290269 | Zbl 0789.03048
[22] Rachůnek, J., Šalounová, D.: State operators on GMV-algebras. Soft Comput. 15 (2011), 327-334. DOI 10.1007/s00500-010-0568-0 | Zbl 1260.06014
[23] Turunen, E.: Boolean deductive systems of BL-algebras. Arch. Math. Logic 40 (2001), 467-473. DOI 10.1007/s001530100088 | MR 1854896 | Zbl 1030.03048
[24] Ward, M., Dilworth, R. P.: Residuated lattices. Trans. Am. Math. Soc. 45 (1939), 335-354. DOI 10.1090/S0002-9947-1939-1501995-3 | MR 1501995 | Zbl 0021.10801
Search
Partner of