On The Spectrum of Countable MV-algebras
محورهای موضوعی : Transactions on Fuzzy Sets and Systems
Giacomo Lenzi
1
(Department of Mathematics, University of Salerno, Fisciano (SA), Italy.)
کلید واژه: MV-algebras, Prime spectrum, Lattice ordered abelian groups.,
چکیده مقاله :
In this paper we consider MV-algebras and their prime spectrum. We show that there is an uncountable MV-algebra that has the same spectrum as the free MV-algebra over one element, that is, the MV-algebra F ree1 of McNaughton functions from [0, 1] to [0, 1], the continuous, piecewise linear functions with integer coefficients. The construction is heavily based on Mundici equivalence between MV-algebras and lattice ordered abelian groups with the strong unit. Also, we heavily use the fact that two MV-algebras have the same spectrum if and only if their lattice of principal ideals is isomorphic. As an intermediate step we consider the MV-algebra A1 of continuous, piecewise linear functions with rational coefficients. It is known that A1 contains F ree1, and that A1 and F ree1 are equispectral. However, A1 is in some sense easy to work with than F ree1. Now, A1 is still countable. To build an equispectral uncountable MV-algebra A2, we consider certain “almost rational” functions on [0, 1], which are rational in every initial segment of [0, 1], but which can have an irrational limit in 1. We exploit heavily, via Mundici equivalence, the properties of divisible lattice ordered abelian groups, which have an additional structure of vector spaces over the rational field.
In this paper we consider MV-algebras and their prime spectrum. We show that there is an uncountable MV-algebra that has the same spectrum as the free MV-algebra over one element, that is, the MV-algebra F ree1 of McNaughton functions from [0, 1] to [0, 1], the continuous, piecewise linear functions with integer coefficients. The construction is heavily based on Mundici equivalence between MV-algebras and lattice ordered abelian groups with the strong unit. Also, we heavily use the fact that two MV-algebras have the same spectrum if and only if their lattice of principal ideals is isomorphic. As an intermediate step we consider the MV-algebra A1 of continuous, piecewise linear functions with rational coefficients. It is known that A1 contains F ree1, and that A1 and F ree1 are equispectral. However, A1 is in some sense easy to work with than F ree1. Now, A1 is still countable. To build an equispectral uncountable MV-algebra A2, we consider certain “almost rational” functions on [0, 1], which are rational in every initial segment of [0, 1], but which can have an irrational limit in 1. We exploit heavily, via Mundici equivalence, the properties of divisible lattice ordered abelian groups, which have an additional structure of vector spaces over the rational field.
[1] Anderson M. and Feil T. Lattice-ordered groups. An introduction. Reidel Texts in the Mathematical Sciences. Dordrecht: D. Reidel Publishing Co.; 1988. x+190. DOI: https://doi.org/10.1007/978-94-009-2871-8
[2] Barbieri G, Di Nola A and Lenzi G. Topological classes of MV-algebras. Submitted.
[3] Belluce L P. Semisimple algebras of infinite-valued logic and bold fuzzy set theory. Canad. J. Math. 1986; 38(6): 1356-1379. DOI: https://doi.org/10.4153/cjm-1986-069-0
[4] Bigard A, Keimel K and Wolfenstein S. Groupes et anneaux réticulés. (French) Lecture Notes in Mathematics, Vol. 608. Berlin-New York: Springer-Verlag; 1977. DOI: https://doi.org/10.1007/bfb0067004
[5] Cignoli R, D’Ottaviano I and Mundici D. Algebraic foundations of many-valued reasoning. Trends in Logic-Studia Logica Library, 7. Dordrecht: Kluwer Academic Publishers; 2000. DOI: https://doi.org/10.1007/978-94-015-9480-6
[6] Di Nola A. Representation and reticulation by quotients of MV-algebras. Ric. Mat. 1991; 40(2): 291-297. DOI: https://doi.org/10.1007/11587.1827-3491
[7] Di Nola A and Lenzi G. The spectrum problem for Abelian ℓ-groups and MV-algebras. Algebra Universalis 2020; 81(39), Paper No. 39, 43 pp. DOI: https://doi.org/10.1007/s00012-020-00668-4
[8] Di Nola A and Leustean I. Lukasiewicz logic and Riesz spaces. Soft Comput. 2014; 18(12): 2349-2363. DOI: https://doi.org/10.1007/s00500-014-1348-z
[9] Mundici D. Advanced Lukasiewicz calculus and MV-algebras. Trends in Logic-Studia Logica Library, 35. Dordrecht: Springer; 2011. https://doi.org/10.1007/978-94-007-0840-2
[10] Mundici D. Interpretation of AF C∗-algebras in Lukasiewicz sentential calculus. J. Funct. Anal. 1986; 65(1): 15-63. DOI: https://doi.org/10.1016/0022-1236(86)90015-7
[11] Stone M. The theory of representations of Boolean algebras. Trans. Amer. Math. Soc. 1936; 40(1): 37-111. DOI: https://doi.org/10.2307/1989664
[12] Stone M. Topological representation of distributive lattices and Brouwerian logics. Casopis Pest. Mat. Fys. 1938; 67(1): 1-25. DOI: https://doi.org/10.21136/cpmf.1938.124080
[13] Wehrung F. Spectral spaces of countable Abelian lattice-ordered groups. Trans. Amer. Math. Soc. 2019; 371(3): 2133-2158. DOI: https://doi.org/10.1090/tran/7596
