zbMATH — the first resource for mathematics

Extending Stone duality to multisets and locally finite MV-algebras. (English) Zbl 1055.06004
A finite {multiset} is a pair $$(X,\sigma)$$ where $$X$$ is finite and where $$\sigma$$ assigns to every $$x\in X$$ a positive natural number, the {multiplicity} of $$x$$. A {morphism} between multisets $$(X,\sigma)$$ and $$(Y,\tau)$$ is a map from $$X$$ to $$Y$$ such that $$\tau(f(x))$$ is a divisor of $$\sigma(x)$$. This makes the finite multisets to a category $$\mathbf{M}$$.
According to the prime factorization theorem, a natural number is uniquely determined by the list of exponents of its prime divisors. A {supernatural} number is defined as a map $$\nu$$ from the ordered list $$P$$ of prime numbers to the set of natural numbers including $$0$$ and $$\infty$$. Every ordinary natural number corresponds naturally to a supernatural number $$\nu_n$$. Pointwise order makes the set $${G}$$ of supernatural numbers $$\nu$$ to a distributive lattice, a fact often used in elementary algebra texts to prove that the natural numbers with respect to divisibility form a distributive lattice. Moreover, $$G$$ is a topological space with the sets $$U_n=\{\nu\mid \nu>\nu_n\}, n=1,2,\ldots\}$$ as open base.
The authors define now the category $$\mathbf{C}$$ where the objects are all pairs $$(S,\sigma)$$ where $$S$$ is a Boolean space and $$\sigma$$ is a continuous map from $$S$$ into $$G$$.
The category $$\mathbf{C}$$ can be construed as a category where the objects are projective limits of the objects of $$\mathbf{M}$$. This goes along the lines of representing an arbitrary Boolean algebra as the direct limit of its finitely generated subalgebras and then dualizing the situation to Stone spaces.
MV-algebras are a non-idempotent generalization of Boolean algebras. The formal definition of these algebras is somewhat technical but they have been around since the work of {C. C. Chang} on multivalued logic in the late fifties.
The authors state as their main result that $$\mathbf{C}$$ and the category of locally finite MV-algebras are co-equivalent.
The paper is clearly written and the organization of the material is exemplary. While most of the material and ideas are probably known to specialists, the reviewer considers nevertheless the paper as a valuable addition to the vast literature on Stone duality.

MSC:
 06D35 MV-algebras 06D50 Lattices and duality
Full Text:
References:
 [1] Aigner, M., Combinatorial theory, (1979), Springer Berlin, Heidelberg, New York · Zbl 0415.05001 [2] Artin, M.; Grothendieck, A.; Verdier, J., Sga 4 (1963-64), Lecture notes in mathematics, Vol. 269, (1972), Springer Berlin, Heidelberg, New York [3] Baer, R., Abelian groups without elements of finite order, Duke math. J, 3, 68-122, (1937) · JFM 63.0074.02 [4] Bigard, A.; Keimel, K.; Wolfenstein, S., Groupes et anneaux Réticulés, Lecture notes in mathematics, Vol. 608, (1977), Springer Berlin, Heidelberg, New York · Zbl 0384.06022 [5] Blizard, W.D., Multiset theory, Notre dame J. formal logic, 30, 36-66, (1989) · Zbl 0668.03027 [6] Burris, S.; Werner, H., Sheaf constructions and their elementary properties, Trans. amer. math. soc, 248, 269-309, (1979) · Zbl 0411.03022 [7] Chang, C.C., Algebraic analysis of many-valued logics, Trans. am. math. soc, 88, 467-490, (1958) · Zbl 0084.00704 [8] Chang, C.C., A new proof of the completeness of the łukasiewicz axioms, Trans. am. math. soc, 93, 74-90, (1959) · Zbl 0093.01104 [9] Cignoli, R.; D’Ottaviano, I.M.; Mundici, D., Algebraic foundations of many-valued reasoning, (2000), Kluwer Dordrecht · Zbl 0937.06009 [10] Cignoli, R.; Dubuc, E.J.; Mundici, D., An MV-algebraic invariant for Boolean algebras with a finite-orbit automorphism, Tatra mount. math. publ, 27, 23-43, (2003) · Zbl 1064.06006 [11] Cignoli, R.; Elliott, G.A.; Mundici, D., Reconstructing C*-algebras from their murray von Neumann orders, Adv. in math, 101, 166-179, (1993) · Zbl 0823.46053 [12] Cignoli, R.; Torrens, A., Boolean products of MV-algebrashypernormal MV-algebras, J. math. anal. appl, 199, 637-653, (1996) · Zbl 0849.06012 [13] Crawley, P.; Dilworth, R.P., Algebraic theory of lattices, (1973), Prentice-Hall Englewood Cliffs, NJ · Zbl 0494.06001 [14] Davey, B., Sheaf spaces and sheaves of universal algebra, Math. zeitschrift, 134, 275-290, (1973) · Zbl 0259.08002 [15] Dixmier, J., On some C★-algebras considered by glimm, J. funct. anal, 1, 122-203, (1967) · Zbl 0152.33003 [16] Effros, E.G., Dimensions and C*-algebras, (1980), American Mathematical Society Providence, RI [17] Fuchs, L., Infinite abelian groups, vol. 2, (1973), Academic Press New York · Zbl 0253.20055 [18] Gierz, G.; Hofmann, K.H.; Keimel, K.; Lawson, J.D.; Mislove, M.; Scott, D.S., A compendium of continuous lattices, (1980), Springer Berlin, Heidelberg, New York · Zbl 0452.06001 [19] Grätzer, G., Universal algebra, (1979), Springer New York [20] Hickman, J.L., A note on the concept of multiset, Bull. aust. math. soc, 22, 211-217, (1980) · Zbl 0432.04005 [21] Jacobson, N., General algebra. II, (1989), Freeman New York [22] Joyal, A.; Tierney, M., An extension of the Galois theory of Grothendieck, Mem. am. math. soc, 309, (1984) · Zbl 0541.18002 [23] K. Keimel, The Representation of Lattice-ordered Groups and Rings by Sections in Sheaves, Lecture Notes in Mathematics, Vol. 248, Springer, Berlin, Heidelberg, New York, 1971, pp. 1-98. [24] Knuth, D.E., The art of computer programming, 2: seminumerical algorithms, (1981), Addison-Wesley Reading, Massachusetts · Zbl 0477.65002 [25] Mac Lane, S., Categories for the working Mathematician, (1998), Springer New York · Zbl 0906.18001 [26] Monro, G.P., The concept of multiset, Z. math. logik grund. math, 33, 171-178, (1987) · Zbl 0609.04008 [27] Mundici, D., Interpretation of AF C* algebras in łukasiewicz sentential calculus, J. funct. anal, 65, 15-63, (1986) · Zbl 0597.46059 [28] J.-P. Serre, Cohomologie Galoisienne. Cours au Collège de France, 1962-1963, Lecture Notes in Mathematics, Vol. 5, Quatrième edition, Springer, Berlin, Heidelberg, New York, 1973. [29] Shatz, S.S., Profinite groups, arithmetic, and geometry, Annals of mathematics studies, Vol. 67, (1972), Princeton University Press Princeton, NJ · Zbl 0236.12002 [30] Simmons, H., A couple of triples, Topol. appl, 13, 201-223, (1982) · Zbl 0484.18005
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.