Previous |  Up |  Next

Article

Keywords:
$n$-valued Łukasiewicz-Moisil algebra; tense $n$-valued Łukasiewicz-Moisil algebra; $n\times m$-valued Łukasiewicz-Moisil algebra
Summary:
In 2000, Figallo and Sanza introduced $n\times m$-valued Łukasiewicz-Moisil algebras which are both particular cases of matrix Łukasiewicz algebras and a generalization of $n$-valued Łukasiewicz-Moisil algebras. Here we initiate an investigation into the class {\bf tLM}$_{n\times m}$ of tense $n\times m$-valued Łukasiewicz-Moisil algebras (or tense LM$_{n\times m}$-algebras), namely $n\times m$-valued Łukasiewicz-Moisil algebras endowed with two unary operations called tense operators. These algebras constitute a generalization of tense Łukasiewicz-Moisil algebras (or tense LM$_{n}$-algebras). Our most important result is a representation theorem for tense LM$_{n\times m}$-algebras. Also, as a corollary of this theorem, we obtain the representation theorem given by Georgescu and Diaconescu in 2007, for tense LM$_{n}$-algebras.
References:
[1] Boicescu, V., Filipoiu, A., Georgescu, G., Rudeanu, S.: Łukasiewicz-Moisil Algebras. Annals of Discrete Mathematics 49 North-Holland, Amsterdam (1991). MR 1112790 | Zbl 0726.06007
[2] Botur, M., Chajda, I., Halaš, R., Kolařík, M.: Tense operators on basic algebras. Int. J. Theor. Phys. 50 (2011), 3737-3749. DOI 10.1007/s10773-011-0748-4 | MR 2860032 | Zbl 1246.81014
[3] Botur, M., Paseka, J.: On tense {MV}-algebras. Fuzzy Sets and Systems 259 (2015), 111-125. MR 3278748 | Zbl 1335.03069
[4] Burgess, J. P.: Basic tense logic. Handbook of Philosophical Logic. Vol. II: Extensions of Classical Logic Synthese Lib. 165 D. Reidel Publishing, Dordrecht (1984), 89-133 D. Gabbay et al. MR 0844597 | Zbl 0875.03046
[5] Chajda, I., Kolařík, M.: Dynamic effect algebras. Math. Slovaca 62 (2012), 379-388. DOI 10.2478/s12175-012-0015-z | MR 2915603 | Zbl 1324.03026
[6] Chajda, I., Paseka, J.: Dynamic effect algebras and their representations. Soft Comput. 16 (2012), 1733-1741. DOI 10.1007/s00500-012-0857-x | Zbl 1318.03059
[7] Chiriţă, C.: Polyadic tense $\theta$-valued Łukasiewicz-Moisil algebras. Soft Comput. 16 (2012), 979-987. DOI 10.1007/s00500-011-0796-y | MR 2760964 | Zbl 1277.03067
[8] Chiriţă, C.: Tense {$\theta$}-valued Łukasiewicz-Moisil algebras. J. Mult.-Val. Log. Soft Comput. 17 (2011), 1-24. MR 2760964 | Zbl 1236.03046
[9] Chiriţă, C.: Tense $\theta$-valued Moisil propositional logic. Int. J. of Computers, Communications and Control 5 (2010), 642-653. DOI 10.15837/ijccc.2010.5.2220
[10] Diaconescu, D., Georgescu, G.: Tense operators on {MV}-algebras and Łukasiewicz-Moisil algebras. Fundam. Inform. 81 (2007), 379-408. MR 2372716 | Zbl 1136.03045
[11] Figallo, A. V., Pelaitay, G.: Discrete duality for tense Łukasiewicz-Moisil algebras. Fund. Inform. 136 (2015), 317-329. MR 3320018 | Zbl 1350.03047
[12] Figallo, A. V., Pelaitay, G.: Tense operators on De Morgan algebras. Log. J. IGPL 22 (2014), 255-267. DOI 10.1093/jigpal/jzt024 | MR 3188082 | Zbl 1347.06012
[13] Figallo, A. V., Pelaitay, G.: Note on tense SH$n$-algebras. An. Univ. Craiova Ser. Mat. Inform. 38 (2011), 24-32. MR 2874020
[14] Figallo, A. V., Pelaitay, G.: Tense operators on SH$n$-algebras. Pioneer J. Algebra Number Theory Appl. 1 (2011), 33-41. MR 3029804
[15] Figallo, A. V., Sanza, C.: Monadic {$n\times m$}-valued Łukasiewicz-Moisil algebras. Math. Bohem. 137 (2012), 425-447. MR 3058274 | Zbl 1274.03104
[16] Figallo, A. V., Sanza, C. A.: The ${\cal NS}_{n\times m}$-propositional calculus. Bull. Sect. Log., Univ. Łód'z, Dep. Log. 37 (2008), 67-79. MR 2460596 | Zbl 1286.03182
[17] Figallo, A. V., Sanza, C.: Álgebras de Łukasiewicz $n\times m$-valuadas con negación. Noticiero Unión Mat. Argent. 93 (2000), 93-94.
[18] Kowalski, T.: Varieties of tense algebras. Rep. Math. Logic 32 (1998), 53-95. MR 1735173 | Zbl 0941.03066
[19] Moisil, G. C.: Essais sur les logiques non chrysippiennes. Éditions de l'Académie de la République Socialiste de Roumanie Bucharest French (1972). MR 0398774 | Zbl 0241.02006
[20] Paseka, J.: Operators on {MV}-algebras and their representations. Fuzzy Sets and Systems 232 (2013), 62-73. DOI 10.1016/j.fss.2013.02.010 | MR 3118535 | Zbl 1314.06016
[21] Sanza, C. A.: On {$n\times m$}-valued Łukasiewicz-Moisil algebras. Cent. Eur. J. Math. 6 (2008), 372-383. DOI 10.2478/s11533-008-0035-7 | MR 2424999 | Zbl 1155.06009
[22] Sanza, C. A.: {$n\times m$}-valued Łukasiewicz algebras with negation. Rep. Math. Logic 40 (2006), 83-106. MR 2207304 | Zbl 1096.03076
[23] Sanza, C.: Álgebras de Łukasiewicz $n\times m$-valuadas con negación. Doctoral Thesis Universidad Nacional del Sur, Bahía Blanca, Argentina (2005).
[24] Sanza, C.: Notes on {$n\times m$}-valued Łukasiewicz algebras with negation. Log. J. IGPL 12 (2004), 499-507. DOI 10.1093/jigpal/12.6.499 | MR 2117684 | Zbl 1062.06018
[25] Suchoń, W.: Matrix Łukasiewicz algebras. Rep. Math. Logic 4 (1975), 91-104. Zbl 0348.02021
Partner of
EuDML logo