Articles de revues : « Lattice-ordered abelian groups »

1

Jakubík, Ján. "Retracts of abelian lattice ordered groups." Czechoslovak Mathematical Journal 39, no. 3 (1989): 477–85. http://dx.doi.org/10.21136/cmj.1989.102319.

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Glass, A. M. W. "Weakly abelian lattice-ordered groups." Proceedings of the American Mathematical Society 129, no. 3 (2000): 677–84. http://dx.doi.org/10.1090/s0002-9939-00-05706-3.

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Glass, A. M. W., Angus Macintyre, and Françoise Point. "Free abelian lattice-ordered groups." Annals of Pure and Applied Logic 134, no. 2-3 (2005): 265–83. http://dx.doi.org/10.1016/j.apal.2004.10.017.

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Conrad, Paul, and J. Roger Teller. "Abelian pseudo lattice ordered groups." Publicationes Mathematicae Debrecen 17, no. 1-4 (2022): 223–41. http://dx.doi.org/10.5486/pmd.1970.17.1-4.26.

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Di Nola, Antonio, Giacomo Lenzi, Gaetano Vitale, and Roberto Giuntini. "Expanding Lattice Ordered Abelian Groups to Riesz Spaces." Mathematica Slovaca 72, no. 1 (2022): 1–10. http://dx.doi.org/10.1515/ms-2022-0001.

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Abstract First we give a necessary and sufficient condition for an abelian lattice ordered group to admit an expansion to a Riesz space (or vector lattice). Then we construct a totally ordered abelian group with two non-isomorphic Riesz space structures, thus improving a previous paper where the example was a non-totally ordered lattice ordered abelian group. This answers a question raised by Conrad in 1975. We give also a partial solution to another problem considered in the same paper. Finally, we apply our results to MV-algebras and Riesz MV-algebras.

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Glass, A. M. W. "Finitely presented ordered groups." Proceedings of the Edinburgh Mathematical Society 33, no. 2 (1990): 299–301. http://dx.doi.org/10.1017/s0013091500018204.

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Ploščica, Miroslav. "Cevian properties in ideal lattices of Abelian ℓ-groups". Forum Mathematicum 33, № 6 (2021): 1651–58. http://dx.doi.org/10.1515/forum-2021-0074.

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Abstract We consider the problem of describing the lattices of compact ℓ {\ell} -ideals of Abelian lattice-ordered groups. (Equivalently, describing the spectral spaces of Abelian lattice-ordered groups.) It is known that these lattices have countably based differences and admit a Cevian operation. Our first result says that these two properties are not sufficient: there are lattices having both countably based differences and Cevian operations, which are not representable by compact ℓ {\ell} -ideals of Abelian lattice-ordered groups. As our second result, we prove that every completely normal distributive lattice of cardinality at most ℵ 1 {\aleph_{1}} admits a Cevian operation. This complements the recent result of F. Wehrung, who constructed a completely normal distributive lattice having countably based differences, of cardinality ℵ 2 {\aleph_{2}} , without a Cevian operation.

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Gluschankof, Daniel, and François Lucas. "Hyper-regular lattice-ordered groups." Journal of Symbolic Logic 58, no. 4 (1993): 1342–58. http://dx.doi.org/10.2307/2275147.

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It is a well-known fact that the notion of an archimedean order cannot be formalized in the first-order calculus. In [12] and [18], A. Robinson and E. Zakon characterized the elementary class generated by all the archimedean, totally-ordered abelian groups (o-groups) in the language 〈+,<〉, calling it the class of regularly ordered or generalized archimedean abelian groups. Since difference (−) and 0 are definable in that language, it is immediate that in the expanded language 〈 +, −, 0, < 〉 the definable expansion of the class of regular groups is also the elementary class generated by the archimedean ones. In the more general context of lattice-ordered groups (l-groups), the notion of being archimedean splits into two different notions: a strong one (being hyperarchimedean) and a weak one (being archimedean). Using the representation theorem of K. Keimel for hyperarchimedean l-groups, we extend in this paper the Robinson and Zakon characterization to the elementary class generated by the prime-projectable, hyperarchimedean l-groups. This characterization is also extended here to the elementary class generated by the prime-projectable and projectable archimedean l-groups (including all complete l-groups). Finally, transferring a result of A. Touraille on the model theory of Boolean algebras with distinguished ideals, we give the classification up to elementary equivalence of the characterized class.We recall that a lattice-ordered group, l-group for short, is a structure

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Glass, A. M. W. "Corrigendum to “Weakly Abelian lattice-ordered groups”." Proceedings of the American Mathematical Society 130, no. 3 (2001): 925–26. http://dx.doi.org/10.1090/s0002-9939-01-06502-9.

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Cignoli, R., D. Gluschankof, and F. Lucas. "Prime spectra of lattice-ordered abelian groups." Journal of Pure and Applied Algebra 136, no. 3 (1999): 217–29. http://dx.doi.org/10.1016/s0022-4049(98)00031-0.

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Elliott, George A., and Daniele Mundici. "A characterisation of lattice-ordered abelian groups." Mathematische Zeitschrift 213, no. 1 (1993): 179–85. http://dx.doi.org/10.1007/bf03025717.

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12

Mundici, Daniele. "Classes of Ultrasimplicial Lattice-Ordered Abelian Groups." Journal of Algebra 213, no. 2 (1999): 596–603. http://dx.doi.org/10.1006/jabr.1998.7679.

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Cherri, Mona, and Wayne B. Powell. "Strong amalgamations of lattice ordered groups and modules." International Journal of Mathematics and Mathematical Sciences 16, no. 1 (1993): 75–80. http://dx.doi.org/10.1155/s0161171293000080.

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We show that every variety of representable lattice ordered groups fails the strong amalgamation property. The same result holds for the variety off-modules over anf-ring. However, strong amalgamations do occur for abelian lattice ordered groups orf-modules when the embeddings are convex.

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14

Kala, Vítězslav. "Lattice-ordered abelian groups finitely generated as semirings." Journal of Commutative Algebra 9, no. 3 (2017): 387–412. http://dx.doi.org/10.1216/jca-2017-9-3-387.

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Jakubík, Ján. "On cut completions of abelian lattice ordered groups." Czechoslovak Mathematical Journal 50, no. 3 (2000): 587–602. http://dx.doi.org/10.1023/a:1022841828655.

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Mundici, Daniele. "Free generating sets of lattice-ordered abelian groups." Journal of Pure and Applied Algebra 211, no. 2 (2007): 400–403. http://dx.doi.org/10.1016/j.jpaa.2007.03.002.

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17

Wehrung, Friedrich. "Spectral spaces of countable Abelian lattice-ordered groups." Transactions of the American Mathematical Society 371, no. 3 (2018): 2133–58. http://dx.doi.org/10.1090/tran/7596.

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18

Pajoohesh, Homeira. "A relationship between the category of chain MV-algebras and a subcategory of abelian groups." Mathematica Slovaca 71, no. 4 (2021): 1027–45. http://dx.doi.org/10.1515/ms-2021-0037.

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Abstract The category of MV-algebras is equivalent to the category of abelian lattice ordered groups with strong units. In this article we introduce the category of circled abelian groups and prove that the category of chain MV-algebras is isomorphic with the category of chain circled abelian groups. In the last section we show that the category of chain MV-algebras is a subcategory of abelian cyclically ordered groups.

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19

CARAMELLO, OLIVIA, and ANNA CARLA RUSSO. "LATTICE-ORDERED ABELIAN GROUPS AND PERFECT MV-ALGEBRAS: A TOPOS-THEORETIC PERSPECTIVE." Bulletin of Symbolic Logic 22, no. 2 (2016): 170–214. http://dx.doi.org/10.1017/bsl.2015.47.

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AbstractWe establish, generalizing Di Nola and Lettieri’s categorical equivalence, a Morita-equivalence between the theory of lattice-ordered abelian groups and that of perfect MV-algebras. Further, after observing that the two theories are not bi-interpretable in the classical sense, we identify, by considering appropriate topos-theoretic invariants on their common classifying topos, three levels of bi-interpretability holding for particular classes of formulas: irreducible formulas, geometric sentences, and imaginaries. Lastly, by investigating the classifying topos of the theory of perfect MV-algebras, we obtain various results on its syntax and semantics also in relation to the cartesian theory of the variety generated by Chang’s MV-algebra, including a concrete representation for the finitely presentable models of the latter theory as finite products of finitely presentable perfect MV-algebras. Among the results established on the way, we mention a Morita-equivalence between the theory of lattice-ordered abelian groups and that of cancellative lattice-ordered abelian monoids with bottom element.

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20

Galli, Adriana, Renato A. Lewin, and Marta Sagastume. "The logic of equilibrium and abelian lattice ordered groups." Archive for Mathematical Logic 43, no. 2 (2004): 141–58. http://dx.doi.org/10.1007/s00153-002-0160-0.

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21

Pap, Endre. "Intrinsic metrics preserving maps on Abelian lattice-ordered groups." Algebra Universalis 29, no. 3 (1992): 338–45. http://dx.doi.org/10.1007/bf01212436.

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22

Scowcroft, Philip. "Algebraically closed and existentially closed Abelian lattice-ordered groups." Algebra universalis 75, no. 3 (2016): 257–300. http://dx.doi.org/10.1007/s00012-016-0375-2.

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23

Cabrer, Leonardo, and Daniele Mundici. "Finitely presented lattice-ordered abelian groups with order-unit." Journal of Algebra 343, no. 1 (2011): 1–10. http://dx.doi.org/10.1016/j.jalgebra.2011.07.007.

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Manara, Corrado, Vincenzo Marra, and Daniele Mundici. "Lattice-ordered Abelian groups and Schauder bases of unimodular fans." Transactions of the American Mathematical Society 359, no. 04 (2006): 1593–605. http://dx.doi.org/10.1090/s0002-9947-06-03935-3.

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25

Ball, Richard N. "Truncated abelian lattice-ordered groups I: The pointed (Yosida) representation." Topology and its Applications 162 (February 2014): 43–65. http://dx.doi.org/10.1016/j.topol.2013.11.007.

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Ball, Richard N. "Truncated abelian lattice-ordered groups II: the pointfree (Madden) representation." Topology and its Applications 178 (December 2014): 56–86. http://dx.doi.org/10.1016/j.topol.2014.08.031.

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27

Gluschankof, Daniel. "The elementary class of products of totally ordered abelian group." Journal of Symbolic Logic 56, no. 1 (1991): 295–99. http://dx.doi.org/10.2307/2274920.

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A basic goal in model-theoretic algebra is to obtain the classification of the complete extensions of a given (first-order) algebraic theory.Results of this type, for the theory of totally ordered abelian groups, were obtained first by A. Robinson and E. Zakon [5] in 1960, later extended by Yu. Gurevich [4] in 1964, and further clarified by P. Schmitt in [6].Within this circle of ideas, we give in this paper an axiomatization of the first-order theory of the class of all direct products of totally ordered abelian groups, construed as lattice-ordered groups (l-groups)—see the theorem below. We think of this result as constituing a first step—undoubtedly only a small one—towards the more general goal of classifying the first-order theory of abelian l-groups.We write groups for abelian l-groups construed as structures in the language 〈 ∨, ∧, +, −, 0〉 (“−” is an unary operation). For unproved statements and unexplicated definitions, the reader is referred to [1].

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Mundici, Daniele. "The Haar theorem for lattice-ordered abelian groups with order-unit." Discrete & Continuous Dynamical Systems - A 21, no. 2 (2008): 537–49. http://dx.doi.org/10.3934/dcds.2008.21.537.

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Marra, Vincenzo. "Lattice-ordered Abelian groups and Schauder bases of unimodular fans, II." Transactions of the American Mathematical Society 365, no. 5 (2013): 2545–68. http://dx.doi.org/10.1090/s0002-9947-2013-05706-6.

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Gurchenkov, S. A., and V. M. Kopytov. "Description of covers of the variety of Abelian lattice-ordered groups." Siberian Mathematical Journal 28, no. 3 (1988): 406–8. http://dx.doi.org/10.1007/bf00969570.

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CABRER, LEONARDO, and DANIELE MUNDICI. "RATIONAL POLYHEDRA AND PROJECTIVE LATTICE-ORDERED ABELIAN GROUPS WITH ORDER UNIT." Communications in Contemporary Mathematics 14, no. 03 (2012): 1250017. http://dx.doi.org/10.1142/s0219199712500174.

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An ℓ-groupG is an abelian group equipped with a translation invariant lattice-order. Baker and Beynon proved that G is finitely generated projective if and only if it is finitely presented. A unital ℓ-group is an ℓ-group G with a distinguished order unit, i.e. an element 0 ≤ u ∈ G whose positive integer multiples eventually dominate every element of G. Unital ℓ-homomorphisms between unital ℓ-groups are group homomorphisms that also preserve the order unit and the lattice structure. A unital ℓ-group (G, u) is projective if whenever ψ : (A, a) → (B, b) is a surjective unital ℓ-homomorphism and ϕ : (G, u) → (B, b) is a unital ℓ-homomorphism, there is a unital ℓ-homomorphism θ : (G, u) → (A, a) such that ϕ = ψ ◦ θ. While every finitely generated projective unital ℓ-group is finitely presented, the converse does not hold in general. Classical algebraic topology (à la Whitehead) is combined in this paper with the Włodarczyk–Morelli solution of the weak Oda conjecture for toric varieties, to describe finitely generated projective unital ℓ-groups.

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32

Jakubik, Jan. "Affine Completeness and Lexicographic Product Decompositions of Abelian Lattice Ordered Groups." Czechoslovak Mathematical Journal 55, no. 4 (2005): 917–22. http://dx.doi.org/10.1007/s10587-005-0075-0.

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33

Kopytov, V. M. "A NON-ABELIAN VARIETY OF LATTICE-ORDERED GROUPS IN WHICH EVERY SOLUBLEl-GROUP IS ABELIAN." Mathematics of the USSR-Sbornik 54, no. 1 (1986): 239–57. http://dx.doi.org/10.1070/sm1986v054n01abeh002969.

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Gluschankof, D. "The Hahn representation theorem for ℓ-groups in ZFA". Journal of Symbolic Logic 65, № 2 (2000): 519–24. http://dx.doi.org/10.2307/2586553.

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In [7] the author discussed the relative force —in the set theory ZF— of some representation theorems for ℓ-groups (lattice-ordered groups). One of the theorems not discussed in that paper is the Hahn representation theorem for abelian ℓ-groups. This result, originally proved by Hahn (see [8]) for totally ordered groups and half a century later by Conrad, Harvey and Holland for the general case (see [4]), states that any abelian ℓ-group can be embedded in a Hahn product of copies of R (the real line with its natural totally-ordered group structure). Both proofs rely heavily on Zorn's Lemma which is equivalent to AC (the axiom of choice).The aim of this work is to point out the use of non-constructible axioms (i.e., AC and weaker forms of it) in the proofs. Working in the frame of ZFA, that is, the Zermelo-Fraenkel set theory where a non-empty set of atoms is allowed, we present alternative proofs which, in the totally ordered case, do not require the use of AC. For basic concepts and notation on ℓ-groups the reader can refer to [1] and [2]. For set theory, to [11].

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35

Reilly, Norman R. "Varieties of lattice ordered groups that contain no non-abelian o-groups are solvable." Order 3, no. 3 (1986): 287–97. http://dx.doi.org/10.1007/bf00400292.

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36

Gehrke, Mai, Samuel J. van Gool, and Vincenzo Marra. "Sheaf representations of MV-algebras and lattice-ordered abelian groups via duality." Journal of Algebra 417 (November 2014): 290–332. http://dx.doi.org/10.1016/j.jalgebra.2014.06.031.

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37

Scowcroft, Philip. "Model-completions for Abelian lattice-ordered groups with finitely many disjoint elements." Annals of Pure and Applied Logic 170, no. 6 (2019): 673–98. http://dx.doi.org/10.1016/j.apal.2019.01.002.

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KAKARIADIS, EVGENIOS T. A. "Finite-dimensional approximations for Nica–Pimsner algebras." Ergodic Theory and Dynamical Systems 40, no. 12 (2019): 3375–402. http://dx.doi.org/10.1017/etds.2019.44.

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We give necessary and sufficient conditions for nuclearity of Cuntz–Nica–Pimsner algebras for a variety of quasi-lattice ordered groups. First we deal with the free abelian lattice case. We use this as a stepping-stone to tackle product systems over quasi-lattices that are controlled by the free abelian lattice and satisfy a minimality property. Our setting accommodates examples like the Baumslag–Solitar lattice for $n=m>0$ and the right-angled Artin groups. More generally, the class of quasi-lattices for which our results apply is closed under taking semi-direct and graph products. In the process we accomplish more. Our arguments tackle Nica–Pimsner algebras that admit a faithful conditional expectation on a small fixed point algebra and a faithful copy of the coefficient algebra. This is the case for CNP-relative quotients in-between the Toeplitz–Nica–Pimsner algebra and the Cuntz–Nica–Pimsner algebra. We complete this study with the relevant results on exactness.

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39

Belluce, L. P., A. Di Nola, and A. Lettieri. "Subalgebras, direct products and associated lattices of MV-algebras." Glasgow Mathematical Journal 34, no. 3 (1992): 301–7. http://dx.doi.org/10.1017/s0017089500008855.

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MV-algebras were introduced by C. C. Chang [3] in 1958 in order to provide an algebraic proof for the completeness theorem of the Lukasiewicz infinite valued propositional logic. In recent years the scope of applications of MV-algebras has been extended to lattice-ordered abelian groups, AF C*-algebras [10] and fuzzy set theory [1].

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40

GLASS, A. M. W., and VINCENZO MARRA. "EMBEDDING FINITELY GENERATED ABELIAN LATTICE-ORDERED GROUPS: HIGMAN'S THEOREM AND A REALISATION OF $\pi$." Journal of the London Mathematical Society 68, no. 03 (2003): 545–62. http://dx.doi.org/10.1112/s002461070300468x.

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41

Jakubík, J. "On the Schröder-Bernstein problem for abelian lattice ordered groups and for MV-algebras." Soft Computing 8, no. 8 (2003): 581–86. http://dx.doi.org/10.1007/s00500-003-0318-7.

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42

Anderson, M., M. Darnel, and T. Feil. "A variety of lattice-ordered groups containing all representable covers of the abelian variety." Order 7, no. 4 (1991): 401–5. http://dx.doi.org/10.1007/bf00383204.

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43

Caramello, O., and A. C. Russo. "The Morita-equivalence between MV-algebras and lattice-ordered abelian groups with strong unit." Journal of Algebra 422 (January 2015): 752–87. http://dx.doi.org/10.1016/j.jalgebra.2014.08.008.

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44

Rump, Wolfgang. "Abelian lattice-ordered groups and a characterization of the maximal spectrum of a Prüfer domain." Journal of Pure and Applied Algebra 218, no. 12 (2014): 2204–17. http://dx.doi.org/10.1016/j.jpaa.2014.03.011.

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45

Burris, Stanley. "A simple proof of the hereditary undecidability of the theory of lattice-ordered abelian groups." Algebra Universalis 20, no. 3 (1985): 400–401. http://dx.doi.org/10.1007/bf01195146.

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46

Glass, A. M. W., Angus Macintyre, and Françoise Point. "Erratum to “Free abelian lattice-ordered groups” [Ann. Pure Appl. Logic 134 (2–3) (2005) 265–283]." Annals of Pure and Applied Logic 167, no. 4 (2016): 431–33. http://dx.doi.org/10.1016/j.apal.2015.11.005.

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Jakubík, Ján. "On the congruence lattice of an abelian lattice ordered group." Mathematica Bohemica 126, no. 3 (2001): 653–60. http://dx.doi.org/10.21136/mb.2001.134195.

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48

Scowcroft, Philip. "Corrigendum to “Model-completions for Abelian lattice-ordered groups with finitely many disjoint elements” [Ann. Pure Appl. Logic 170 (2019) 673–698]." Annals of Pure and Applied Logic 170, no. 11 (2019): 102720. http://dx.doi.org/10.1016/j.apal.2019.102720.

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Harminc, Matúš. "Cardinality of the system of all sequential convergences on an Abelian lattice ordered group." Czechoslovak Mathematical Journal 37, no. 4 (1987): 533–46. http://dx.doi.org/10.21136/cmj.1987.102181.

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CABRER, LEONARDO MANUEL, and DANIELE MUNDICI. "Classifying orbits of the affine group over the integers." Ergodic Theory and Dynamical Systems 37, no. 2 (2015): 440–53. http://dx.doi.org/10.1017/etds.2015.45.

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For each $n=1,2,\ldots ,$ let $\text{GL}(n,\mathbb{Z})\ltimes \mathbb{Z}^{n}$ be the affine group over the integers. For every point $x=(x_{1},\ldots ,x_{n})\in \mathbb{R}^{n}$ let $\text{orb}(x)=\{\unicode[STIX]{x1D6FE}(x)\in \mathbb{R}^{n}\mid \unicode[STIX]{x1D6FE}\in \text{GL}(n,\mathbb{Z})\ltimes \mathbb{Z}^{n}\}.$ Let $G_{x}$ be the subgroup of the additive group $\mathbb{R}$ generated by $x_{1},\ldots ,x_{n},1$. If $\text{rank}(G_{x})\neq n$ then $\text{orb}(x)=\{y\in \mathbb{R}^{n}\mid G_{y}=G_{x}\}$. Thus, $G_{x}$ is a complete classifier of $\text{orb}(x)$. By contrast, if $\text{rank}(G_{x})=n$, knowledge of $G_{x}$ alone is not sufficient in general to uniquely recover $\text{orb}(x)$; as a matter of fact, $G_{x}$ determines precisely $\max (1,\unicode[STIX]{x1D719}(d)/2)$ different orbits, where $d$ is the denominator of the smallest positive non-zero rational in $G_{x}$ and $\unicode[STIX]{x1D719}$ is the Euler function. To get a complete classification, rational polyhedral geometry provides an integer $1\leq c_{x}\leq \max (1,d/2)$ such that $\text{orb}(y)=\text{orb}(x)$ if and only if $(G_{x},c_{x})=(G_{y},c_{y})$. Applications are given to lattice-ordered abelian groups with strong unit and to AF $C^{\ast }$-algebras.

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