Some classical theorems in state residuated lattices

Document Type : Research Paper

Authors

1 Department of Mathematics, Shahrekord Branch, Islamic Azad University, Shahrekord, Iran.

2 Department of Mathematics, Persian Gulf University, Bushehr, Iran.

10.29252/as.2020.1910

Abstract

This paper, by considering the notion of a state residuated lattice morphism in the class of state residuated lattices, investigates some classical theorems namely the going up and lying over theorems. Results show that each state residuated lattice morphism fulfills these theorems. Also, some properties about prime filters of residuated lattices are obtained which are given in the paper.

Keywords


[1] L.P. Belluce, The going up and going down theorems in MV-algebras and abelian groups, Journal of mathematical analysis and applications. Vol. 241 No. 4 (2000), pp. 92–106.
[2] L.C. Ciungu, A. Dvureˇcenskij and M. Hyˇcko, State BL-algebras, Soft Computing Vol. 15 No. 4 (2011), pp. 619–634.
[3] N. Constantinescu, On pseudo BL-algebras with internal state, Soft Computing. Vol. 16 No. 11 (2012), pp. 1915–1922.
[4] Z. Dehghani and F. Forouzesh, State filters in state residuated lattices, Categories and General Algebraic Structures with Applications. Vol. 10 No. 1 (2019), pp. 17–37.
[5] A. Di Nola and A. Dvurecenskij, On some classes of state-morphism MValgebras, Mathematica Slovaca. Vol. 59 No. 5 (2009a), pp. 517–534.
[6] A. Di Nola and A. Dvurecenskij, State-morphism MV-algebras, Annals of Pure and Applied Logic. Vol. 161 No. 2 (2009b), pp. 161–173.
[7] A. Dvurecenskij, J. Rachonek and D. Salounove,  State operators on generalizations of fuzzy structures, Fuzzy sets and systems. Vol. 187 No. 1 (2012), pp. 58–76.
[8] T. Flaminio and F. Montagna, An algebraic approach to states on MV-algebras, in ‘EUSFLAT Conf.(2) (2007), pp. 201–206.
[9] T. Flaminio and F. Montagna, MV-algebras with internal states and probabilistic fuzzy logics, International Journal of Approximate Reasoning. Vol. 50 No. 1 (2009), pp. 138–152.
[10] N. Galatos, P. Jipsen and T. Kowalski, Residuated lattices: an algebraic glimpse at substructural logics, Vol. 151, Elsevier, (2007). [11] G. Georgescu and C. Muresan, Going up and lying over in congruence-modular algebras, Mathematica Slovaca. Vol. 69 No. 2 (2019), pp. 275–296.
[12] P. He, X. Xin and Y. Yang, On state residuated lattices, Soft Computing. Vol. 19 No. 8 (2015), pp. 2083– 2094.
[13] P. Jipsen and C. Tsinakis, A survey of residuated lattices in Ordered algebraic structures, Springer, (2002), pp. 19–56.
[14] M. Kondo, Generalized state operators on residuated lattices, Soft Computing. Vol. 21 No. 20 (2017), pp. 6063–6071.
[15] M. Kondo and M.F. Kawaguchi Some properties of generalized state operators on residuated lattices, IEEE 46th International Symposium on Multiple-Valued Logic (ISMVL)’, IEEE, (2016) pp. 162–166.
[16] S. Rasouli, The going-up and going-down theorems in residuated lattices, Soft Computing. Vol. 23 No. 17 (2017), pp. 7621–7635.
[17] S. Rasouli and B. Avvaz An investigation on boolean prime lters in BL-algebras, Soft Computing. Vol. 19 No. 10 (2015), pp. 2743–2750.
[18] S. Rasouli and S. Zarin On residuated lattices with left and right internal state, Fuzzy Sets and Systems. Vol. 373 (2019), pp. 37–61.
[19] M. Taheri, F. Khaksar Haghani and S. Rasouli, Simple, local and subdirectly irreducible state residuated lattices, (accepted by Revista de la union matematica argrntina on December 17, 2019)