Artículo
Abstract:
In joint work with E. Dubuc and D. Mundici, the first author extended Stone duality for boolean algebras to locally finite MV-algebras. On the topological side of the duality, one has to assign to each point of a boolean space a generalized natural number by way of a multiplicity, so as to make the assignment continuous with respect to the Scott topology of the lattice of generalized natural numbers. The continuous maps between such generalized multisets are to be multiplicity-decreasing with respect to the divisibility order of generalized natural numbers. In this paper we extend these results to the class of MV-algebras that are locally weakly finite, i.e., such that all their finitely generated subalgebras split into a finite direct product of simple MV-algebras. Using the Scott topology on the lattice of subalgebras of the real unit interval [0, 1] (regarded with its natural MV-algebraic structure), we construct a 'real-valued multiset' over the (boolean) space of maximal ideals of a locally weakly finite MV-algebra. Building on this, we obtain a duality for locally weakly finite MV-algebras that includes as a special case the above-mentioned duality for locally finite MV-algebras. We give an example that shows that the duality established in this paper via the Scott topology cannot be extended, without non-trivial modifications, to larger classes of algebras. © de Gruyter 2012.
Registro:
| Documento: | Artículo |
| Título: | Stone duality for real-valued multisets |
| Autor: | Cignoli, R.; Marra, V. |
| Filiación: | Departamento de Matemótica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, 1428 Buenos Aires, Argentina Dipartimento di Informatica e Comunicazione, Università degli Studi di Milano, via Comelico 39-41, 20135 Milano, Italy |
| Palabras clave: | Algebraic lattices; Global sections; Global sheaves; Multisets; MV-algebras; Scott topology |
| Año: | 2012 |
| Volumen: | 24 |
| Número: | 6 |
| Página de inicio: | 1317 |
| Página de fin: | 1331 |
| DOI: | http://dx.doi.org/10.1515/FORM.2011.109 |
| Título revista: | Forum Mathematicum |
| Título revista abreviado: | Forum Math. |
| ISSN: | 09337741 |
| Registro: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_09337741_v24_n6_p1317_Cignoli |
Referencias:
- Bigard, A., Keimel, K., Wolfenstein, S., Groupes et Anneaux Réticulés (1977) Lecture Notes in Mathematics, 608. , Springer-Verlag, Berlin, Heidelberg, New York
- Borkowski, L., (1970) Selected Works of J. Łukasiewicz, , North-Holland, Amsterdam
- Bosbach, B., Concerning bricks (1981) Acta Math. Hungar., 38, pp. 89-104
- Bosbach, B., Residuated grupoids - Again (2009) Results Math., 53, pp. 27-51
- Burris, S., Sankappanavar, H.P., (1981) A Course in Universal Algebra, , Springer-Verlag, New York
- Chang, C.C., Algebraic analysis of many-valued logics (1958) Trans. Amer. Math. Soc., 88, pp. 467-490
- Chang, C.C., A new proof of the completeness of the Łukasiewicz axioms (1959) Trans. Amer. Math. Soc., 93, pp. 74-90
- Cignoli, R., D'Ottaviano, I.M., Mundici, D., (2000) Algebraic Foundations of Many-Valued Reasoning, Trends in Logic, 7. , Kluwer, Dordrecht, Boston, London
- Cignoli, R., Dubuc, E.J., Mundici, D., Extending stone duality to multisets and locally finite MV-algebras (2004) J. Pure Appl. Algebra, 189, pp. 37-59
- Dubuc, E.J., Proveda, Y., Representation theory of MV-algebras (2010) Ann. Pure Appl. Logic, 161, pp. 1024-1046
- Effros, E.G., (1980) Dimensions and C * -Algebras, , American Mathematical Society, Providence, RI
- Filipoiu, A., Georgescu, G., Lettieri, A., Maximal MV-algebras (1997) Mathware Soft Comput., 4, pp. 53-62
- Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M., Scott, D.S., (1980) A Compendium of Continuous Lattices, , Springer-Verlag, Berlin
- Gispert, J., Mundici, D., MV-algebras: A variety for magnitudes with archimedean units (2005) Algebra Universalis, pp. 7-43
- Iséki, K., Tanaka, S., An introduction to the theory of BCK-algebras (1978) Math. Japon., 23, pp. 1-26
- Łukasiewicz, J., Tarski, A., Untersuchungen über den Aussagenkalkül (1930) Comptes Rendus de la Société des Sciences et des Lettres de Varsovie, 23, pp. 30-50. , (English translation in [2] and [23])
- Marra, V., Mundici, D., Łukasiewicz logic and chang's MV algebras in action (2003) 50 Years of Studia Logica, pp. 145-192. , edited by V. Hendricks and J. Malinowski, Trends in Logic 21, Kluwer, Dordrecht
- Mundici, D., Interpretation of AF C * algebras in Łukasiewicz sentential calculus (1986) J. Appl. Funct. Anal., 65, pp. 15-63
- Mundici, D., MV-algebras are categorically equivalent to bounded commutative BCK-algebras (1986) Math. Japon., 31, pp. 889-894
- Mundici, D., Advanced Łukasiewicz calculus and MV-algebras (2011) Trends in Logic, 35. , Springer-Verlag, New York
- Stone, M.H., The theory of representations for Boolean algebras (1936) Trans. Amer. Math. Soc., 40, pp. 37-111
- Stone, M.H., Applications of the theory of Boolean rings to general topology (1937) Trans. Amer. Math. Soc., 41, pp. 375-481
- Tarski, A., (1983) Logic, Semantics, Metamathematics, , Clarendon Press, Oxford, 1956. Reprinted Hackett, Indianapolis
Citas:
---------- APA ----------
Cignoli, R. & Marra, V. (2012) . Stone duality for real-valued multisets. Forum Mathematicum, 24(6), 1317-1331.http://dx.doi.org/10.1515/FORM.2011.109
---------- CHICAGO ----------
Cignoli, R., Marra, V. "Stone duality for real-valued multisets" . Forum Mathematicum 24, no. 6 (2012) : 1317-1331.http://dx.doi.org/10.1515/FORM.2011.109
---------- MLA ----------
Cignoli, R., Marra, V. "Stone duality for real-valued multisets" . Forum Mathematicum, vol. 24, no. 6, 2012, pp. 1317-1331.http://dx.doi.org/10.1515/FORM.2011.109
---------- VANCOUVER ----------
Cignoli, R., Marra, V. Stone duality for real-valued multisets. Forum Math. 2012;24(6):1317-1331.http://dx.doi.org/10.1515/FORM.2011.109
