State filters in state residuated lattices

Document Type : Research Paper


1 Higher Education Complex of Bam, Iran

2 Faculty of Mathematics and computing, Higher Education Complex of Bam, Kerman, Iran.



In this paper, we introduce the notions of prime state filters, obstinate state filters, and primary state filters in state residuated lattices and study some properties of them. Several characterizations of these state filters are given and the prime state filter theorem is proved. In addition, we investigate the relations between them.


[1] Balbes, R. and Dwinger, P., “Distributive lattices”, XIII. University of Missouri Press, 1974.
[2] Borumand Saeid, A. and Pourkhatoun, M., Obstinate filters in residuated lattices, Bull. Math. Soc. Sci. Math. Roumanie, Nouvelle Série 55 (103)(4) (2012) 413-422.
[3] Ciungu, L.C., Bosbach and Rieˇcan states on residuated lattices, J. Appl. Funct. Anal. 3(1) (2008), 175-188.
[4] Dvureˇcenskij, A., States on pseudo MV-algebras, Studia Logica 68 (2001), 301-327.
[5] Forouzesh, F., Eslami, E., and Borumand Saeid, A., On obstinate ideals in MV-Algebras, U.P.B. Sci. Bull., Series A, 76(2) (2014), 53-62.
[6] Georgescu, G., Bosbach states on fuzzy structures, Soft Comput. 8 (2004), 217-230.
[7] Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M., and Scott, D.S., “Continuous Lattices and Domains”, Cambridge University Press, 2003.
[8] Gratzer, G., “Lattice theory”, First Concepts and Distributive Lattices, A Series of Books in Mathematics, W.H. Freeman and Company, 1972.
[9] Hajek, P., “Metamathematics of Fuzzy Logic”, Trends in Logic Studia Logica Library 4, Kluwer Academic Publishers, 1998.
[10] He, P., Xin, X., and Yang, Y., On state residuated lattices, Soft Comput. 19 (2015), 2083-2094.
[11] Kroupa, T., Every state on semisimple MV-algebra is integral, Fuzzy Sets and Systems 157 (2006), 2771-2782.
[12] Liu, L. and Li, K., Boolean filters and positive implicative filters of residuated lattices, Inf. Sci. 177 (2007), 5725-5738.
[13] Liu, L.Z. and Zhang, X.Y., States on finite linearly ordered IMT L-algebras, Soft Comput. 15 (2011), 2021-2028.
[14] Liu, L.Z. and Zhang, X.Y., States on R0-algebras, Soft Comput. 12 (2008), 1099-1104.
[15] Liu, L.Z., On the existence of states on MT L-algebras, Inf. Sci. 220 (2013), 559-567.
[16] Mundici, D., Averaging the truth-value in Łukasiewicz sentential logic, Studia Logica 55 (1995), 113-127.
[17] Muresan, C., Dense elements and classes of residuated lattices, Bull. Math. Soc. Sci. Math. Roumanie. 53 (2010), 11-24.
[18] Piciu, D., “Algebras of Fuzzy Logic”. Ed. Universitaria, 2007.
[19] Rieˇcan, B., On the probability on BL-algebras, Acta Math. Nitra 4 (2000), 3-13.
[20] Turunen, E. and Mertanen, J., States on semi-divisible residuated lattices, Soft Comput. 12 (2008), 353-357.
[21] Turunen, E. “Mathematics Behind Fuzzy Logic”, Advances in Soft Computing, Physica-Verlag, 1999.
[22] Gasse, B. Van., Deschrijver, G., Cornelis, C., and Kerre, E.E., Filters of residuated lattices and triangle algebras, Inform. Sci. 180 (2010), 3006-3020.
[23] Ward, M. and Dilworth, P.R., Residuated lattice, Trans Am. Math. Soc. 45 (1939), 335-354.