# zbMATH — the first resource for mathematics

Operators on MV-algebras and their representations. (English) Zbl 1314.06016
Summary: A crucial problem concerning (tense) operators on MV-algebras is their representations. Having an MV-algebra with (tense) operators, we can ask if there exists a frame yielding a representation of this MV-algebra. We solve this problem for semisimple MV-algebras.

##### MSC:
 06D35 MV-algebras 03G25 Other algebras related to logic
Full Text:
##### References:
 [1] Belluce, L. P., Semisimple algebras of infinite-valued logic and bold fuzzy set theory, Can. J. Math., 38, 1356-1379, (1986) · Zbl 0625.03009 [2] Belluce, L. P.; Di Nola, A.; Lenzi, G., Hyperfinite MV-algebras, J. Pure Appl. Algebra, 217, 1208-1223, (2013) · Zbl 1276.06006 [3] M. Botur, J. Paseka, Tense MV-algebras, Archive for Mathematical Logic, accepted 2012. · Zbl 1335.03069 [4] M. Botur, J. Paseka, On the extensions of Di Nola’s Theorem, preprint 2012. · Zbl 1320.06010 [5] Burges, J., Basic tense logic, (Gabbay, D. M.; Günther, F., Handbook of Philosophical Logic, vol. 1II, (1984), D. Reidel Publishing Company), 89-139 [6] Butnariu, D.; Klement, E. P., Triangular norm-based measures and games with fuzzy coalitions, (1993), Kluwer Academic Publishers Dordrecht · Zbl 0804.90145 [7] Chang, C. C., Algebraic analysis of many-valued logics, Trans. Amer. Math. Soc., 88, 467-490, (1958) · Zbl 0084.00704 [8] Cignoli, R.; D’Ottaviano, I.; Mundici, D., Algebraic Foundations of Many-valued Reasoning, Trends in Logic, vol. 7, (2000), Kluwer Academic Publishers [9] Diaconescu, D.; Georgescu, G., Tense operators on MV-algebras and łukasiewicz-moisil algebras, Fund. Inf., 81, 379-408, (2007) · Zbl 1136.03045 [10] Diaconescu, D.; Georgescu, G., Forcing operators on MTL-algebras, Math. Log. Quart., 57, 47-64, (2011) · Zbl 1215.03075 [11] Di Nola, A., Representation and reticulation by quotients of MV-algebras, Ricerche di Mat., XL, 291-297, (1991) · Zbl 0767.06013 [12] Di Nola, A.; Navara, M., The $$\sigma$$-complete MV-algebras which have enough states, Colloq. Math., 103, 121-130, (2005) · Zbl 1081.06011 [13] Georgescu, G.; Leustean, I., A representation theorem for monadic pavelka algebras, J. Universal Comput. Sci., 6, 105-111, (2000) · Zbl 0963.03088 [14] G. Hansoul, B. Teheux, Completeness results for many-valued Łukasiewicz modal systems and relational semantics, 2006. Available at $$\langle$$http://arxiv.org/abs/math/0612542$$\rangle$$. [15] Łukasiewicz, J., On three-valued logic, (Borkowski, L., Selected Works by Jan Łukasiewicz, (1970), North-Holland Amsterdam), 87-88 [16] Ostermann, P., Many-valued modal propositional calculi, Z. Math. Logik Grundlag. Math., 34, 343-354, (1988) · Zbl 0661.03011 [17] Teheux, B., A duality for the algebras of a łukasiewicz $$n + 1$$-valued modal system, Studia Logica, 87, 13-36, (2007) · Zbl 1127.03050 [18] B. Teheux, Algebraic Approach to Modal Extensions of Łukasiewicz Logics, Doctoral Thesis, Université de Liege, 2009 $$\langle$$http://orbi.ulg.ac.be/handle/2268/10887$$\rangle$$.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.