×

zbMATH — the first resource for mathematics

Idempotent endomorphisms of free MV-algebras and unital \(\ell\)-groups. (English) Zbl 1362.06003
For counting the number of idempotent endomorphisms of a structure \(S\) onto a substructure \(T\), the authors consider the case when \(T\) is a projective lattice-ordered abelian group with a distinguished strong order unit, or equivalently, a projective MV-algebra. Suppose \(A\) is the image (= range) of an idempotent endomorphism of the free \(n\)-generator MV-algebra \(\mathcal M ([0,1]^n)\) of McNaughton functions on \([0,1]^n\). They prove that the number \(\mathrm r(A)\) of idempotent endomorphisms of \(\mathcal M([0,1]^n)\) onto \(A\) is finite if, and only if, the maximal spectral space \(\mu_A\) is homeomorphic to a (Kuratowski) closed domain \(M\) of \([0,1]^n\), in the sense that \(M=\mathrm{cl}(\operatorname{int}(M))\). Further, the closed domain condition is decidable and \(\mathrm r(A)\) is computable, once an idempotent endomorphism of \(\mathcal M([0,1]^n)\) onto \(A\) is explicitly given. Thus, every finitely generated projective MV-algebra \(B\) comes equipped with a new invariant \(\iota(B)=\sup\{\mathrm r(A)\mid A\cong B\), for \(A\) the image of an idempotent endomorphism of \(\mathcal M([0,1]^k)\}\), and \(k\) the smallest number of generators of \(B\). We compute \(\iota(B)\) for many projective MV-algebras \(B\) existing in the literature. Various problems concerning idempotent endomorphisms of free MV-algebras are shown to be decidable.
MSC:
06D35 MV-algebras
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
08B30 Injectives, projectives
52B70 Polyhedral manifolds
55U10 Simplicial sets and complexes in algebraic topology
57Q05 General topology of complexes
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Baker, K. A., Free vector lattices, Can. J. Math., 20, 58-66, (1968) · Zbl 0157.43401
[2] Beynon, W. M., On rational subdivisions of polyhedra with rational vertices, Can. J. Math., 29, 238-242, (1977) · Zbl 0343.52006
[3] Beynon, W. M., Applications of duality in the theory of finitely generated lattice-ordered abelian groups, Can. J. Math., 29, 243-254, (1977) · Zbl 0361.06017
[4] (Bilen Can, M.; Li, Z.; Steinberg, B.; Wang, Q., Algebraic Monoids, Group Embeddings, and Algebraic Combinatorics, (2014), Springer)
[5] Byrd, R. D.; Lloyd, J. T.; Mena, R. A., On the retractability of some one-relator groups, Pac. J. Math., 72, 351-359, (1977) · Zbl 0344.20030
[6] Cabrer, L., Simplicial geometry of unital lattice-ordered abelian groups, Forum Math., 27, 3, 1309-1344, (2015) · Zbl 1330.06013
[7] Cabrer, L., Rational simplicial geometry and projective lattice-ordered abelian groups, (28 May 2014)
[8] Cabrer, L.; Mundici, D., Projective MV-algebras and rational polyhedra, Algebra Univers., 62, 63-74, (2009), special issue in memoriam Paul Conrad (J. Martínez, et al., Eds.) · Zbl 1196.06004
[9] Cabrer, L.; Mundici, D., Rational polyhedra and projective lattice-ordered abelian groups with order unit, Commun. Contemp. Math., 14, 3, 1250017, (2012) · Zbl 1275.06002
[10] Cabrer, L. M.; Mundici, D., A stone-Weierstrass theorem for MV-algebras and unital -groups, J. Log. Comput., 25, 3, 683-699, (2015) · Zbl 1323.06011
[11] Cabrer, L. M.; Mundici, D., Classifying orbits of the affine group over the integers, Ergod. Theory Dyn. Syst., (2016), in press. Published online 22 July 2015
[12] Cabrer, L. M.; Spada, L., MV-algebras, infinite dimensional polyhedra and natural dualities, available at · Zbl 1367.06006
[13] Caramello, O.; Russo, A. C., The Morita-equivalence between MV-algebras and lattice-ordered abelian groups with strong unit, J. Algebra, 422, 752-787, (2015) · Zbl 1320.06011
[14] Cignoli, R.; D’Ottaviano, I. M.L.; Mundici, D., Algebraic foundations of many-valued reasoning, Trends Log., vol. 7, (2000), Kluwer Dordrecht · Zbl 0937.06009
[15] Di Nola, A.; Grigolia, R.; Panti, G., Finitely generated free MV-algebras and their automorphism groups, Stud. Log., 61, 65-78, (1998) · Zbl 0964.06010
[16] Dubuc, E.; Poveda, Y., Representation theory of MV-algebras, Ann. Pure Appl. Logic, 161, 1024-1046, (2010) · Zbl 1229.06006
[17] Engelking, R., General topology, Sigma Ser. Pure Math., vol. 6, (1989), Herldermann Verlag Berlin · Zbl 0684.54001
[18] Evans, D. M., Model theory of groups and automorphism groups, (1997), Cambridge University Press · Zbl 0878.03026
[19] Ewald, G., Combinatorial convexity and algebraic geometry, Grad. Texts Math., vol. 168, (1996), Springer-Verlag Berlin, Heidelberg · Zbl 0869.52001
[20] Fuchs, L., Note on the construction of free MV-algebras, Algebra Univers., 62, 45-49, (2009) · Zbl 1196.06006
[21] Glass, A. M.W.; Madden, J. J., The word problem versus the isomorphism problem, J. Lond. Math. Soc., 30, 53-61, (1984) · Zbl 0551.20018
[22] Johnstone, P. T., Sketches of an elephant: A topos theory compendium, vol. 2, (2002), Clarendon Press Oxford · Zbl 1071.18001
[23] Kolařík, M., Independence of the axiomatic system for MV-algebras, Math. Slovaca, 63, 1-4, (2013) · Zbl 1313.06017
[24] Kubials, W.; Uspenskij, V., A compact group which is not Valdivia compact, Proc. Am. Math. Soc., 133, 8, 2483-2487, (1977) · Zbl 1066.54027
[25] Mac Lane, S., Categories for the working Mathematician, Grad. Texts Math., vol. 5, (1998), Springer · Zbl 0906.18001
[26] Marra, V.; Mundici, D., The Lebesgue state of a unital abelian lattice-ordered group, J. Group Theory, 10, 655-684, (2007) · Zbl 1136.06009
[27] Marra, V.; Spada, L., Duality, projectivity, and unification in łukasiewicz logic and MV-algebras, Ann. Pure Appl. Logic, 164, 192-210, (2013) · Zbl 1275.03099
[28] Mundici, D., Interpretation of AF \(C^\ast\)-algebras in łukasiewicz sentential calculus, J. Funct. Anal., 65, 15-63, (1986) · Zbl 0597.46059
[29] Mundici, D., The Haar theorem for lattice-ordered abelian groups with order-unit, Discrete Contin. Dyn. Syst., 21, 537-549, (2008) · Zbl 1154.28007
[30] Mundici, D., Advanced łukasiewicz calculus and MV-algebras, Trends Log., vol. 35, (2011), Springer Berlin · Zbl 1235.03002
[31] Mundici, D., Invariant measure under the affine group over Z, Comb. Probab. Comput., 23, 248-268, (2014) · Zbl 1298.52016
[32] Panti, G., Multi-valued logics, (Smets, P., Quantified Representation of Uncertainty and Imprecision, (1998), Kluwer Academic Publishers Dordrecht), 25-74 · Zbl 0929.03033
[33] Schein, B. M.; Teclezghi, B., Endomorphisms of finite full transformation semigroups, Proc. Am. Math. Soc., 126, 9, 2579-2587, (1998) · Zbl 0912.20047
[34] Shtan’ko, M. A., Markov’s theorem and algorithmically non-recognizable combinatorial manifolds, Izv. Math., 68, 207-224, (2004)
[35] Stillwell, J., Classical topology and combinatorial group theory, Grad. Texts Math., vol. 72, (1980), Springer NY · Zbl 0453.57001
[36] Waterhouse, W. C., Retractions of separable commutative algebras, Arch. Math. (Basel), 60, 36-39, (1993) · Zbl 0806.16016
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.