F–multipliers and the localization of LMn×m–algebras

Gallardo, C.; Sanza, C.; Ziliani, A.

doi:10.2478/auom-2013-0019

Abstract

In this note, we introduce the notion of n × m-ideal on n × m- valued Łukasiewicz-Moisil algebras (or LM_n×m-algebras) which allows us to consider a topology on them. Besides, we define the concept of F-multiplier, where F is a topology on an LM_n×m-algebra L, which is used to construct the localization LM_n×m-algebra L_F. Furthermore, we prove that the LM_n×m-algebra of fractions L_Sassociated with an ⋀-closed subset S of L is an LM_n×m-algebra of localization. Finally, in the finite case we prove that L_Sis isomorphic to a special subalgebra of L. Since n-valued Łukasiewicz-Moisil algebras are a particular case of LM_n×m-algebras, all these results generalize those obtained in [4] (see also [3]).

References

[1] V. Boicescu, A. Filipoiu, G. Georgescu and S. Rudeanu, Lukasiewicz- Moisil Algebras, North-Holland, Amsterdam, 1991.
Search in Google Scholar
[2] A. Brezuleanu and R. Diaconescu, Sur la duale de la cat´egorie des treillis, Rev. Roum. Math. Pures et Appl., XIV, 3, (1969), 311-323.
Search in Google Scholar
[3] D. Bu¸sneag and F. Chirte¸s, LMn-algebra of fractions and maximal LMn- algebra of fractions. Discrete Math. 296 (2005), 2-3, 143-165.
Search in Google Scholar
[4] F. Chirte¸s, Localization of LMn-algebras, Cent. Eur. J. Math. 3 (2005), 1, 105-124.10.2478/BF02475659
Search in Google Scholar
[5] W. Cornish, The multiplier extension of a distributive lattice, J. Algebra, 32, (1974), 339-355.10.1016/0021-8693(74)90143-4
Search in Google Scholar
[6] A.V. Figallo and C. Sanza, Algebras de Lukasiewicz n ×m-valuadas con negaci´on, Noticiero de la Uni´on Matemática Argentina (2000), 93.
Search in Google Scholar
[7] A.V. Figallo and C. Sanza, The NSn×m-propositional calculus. Bull. Sect. Logic Univ. L´odz 37 (2008), 2, 67-79.
Search in Google Scholar
[8] G. Georgescu, F-multipliers and the localization of distributive lattices, Algebra Universalis, 21 (1985), 181-197.10.1007/BF01188055
Search in Google Scholar
[9] Gr . C. Moisil, Essais sur les Logiques non Chrysippiennes, Bucarest, 1972.
Search in Google Scholar
[10] N. Popescu, Abelian categories with applications to rings and modules. Academic Press, 1973.
Search in Google Scholar
[11] S. Rudeanu, Localizations and fractions in algebra of logic, J. of Mult.- Valued Logic and Soft Computing, 16 (2010), 467-504.
Search in Google Scholar
[12] C. Sanza, Notes on n × m-valued Lukasiewicz algebras with negation, Logic Journal of the IGPL 6, 12 (2004), 499-507.10.1093/jigpal/12.6.499
Search in Google Scholar
[13] C. Sanza, n × m-valued Lukasiewicz algebras with negation, Reports on Mathematical Logic 40 (2006), 83-106.
Search in Google Scholar
[14] C. Sanza, On n × m-valued Lukasiewicz-Moisil algebras, Centr. Eur. J. Math. 6, 3 (2008), 372-383.10.2478/s11533-008-0035-7
Search in Google Scholar
[15] B. Stenstr¨om, Rings of quotients, Springer-Verlag, 1975.10.1007/978-3-642-66066-5
Search in Google Scholar
[16] W. Sucho´n, Matrix Lukasiewicz Algebras, Reports on Mathematical Logic 4 (1975), 91-104.
Search in Google Scholar

F–multipliers and the localization of LMn×m–algebras

Abstract

Paradigm

My account