Journal articles: '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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2

Glass, A. M. W. "Weakly abelian lattice-ordered groups." Proceedings of the American Mathematical Society 129, no. 3 (September 20, 2000): 677–84. http://dx.doi.org/10.1090/s0002-9939-00-05706-3.

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3

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

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4

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

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5

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

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6

Di Nola, Antonio, Giacomo Lenzi, Gaetano Vitale, and Roberto Giuntini. "Expanding Lattice Ordered Abelian Groups to Riesz Spaces." Mathematica Slovaca 72, no. 1 (February 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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7

GÖBEL, RÜDIGER, and SAHARON SHELAH. "CHARACTERIZING AUTOMORPHISM GROUPS OF ORDERED ABELIAN GROUPS." Bulletin of the London Mathematical Society 35, no. 03 (May 2003): 289–92. http://dx.doi.org/10.1112/s0024609302001881.

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8

Goffman. "COMPLETENESS IN TOTALLY ORDERED ABELIAN GROUPS." Real Analysis Exchange 20, no. 1 (1994): 58. http://dx.doi.org/10.2307/44152461.

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9

Dolich, Alfred, and John Goodrick. "Strong theories of ordered Abelian groups." Fundamenta Mathematicae 236, no. 3 (2017): 269–96. http://dx.doi.org/10.4064/fm256-5-2016.

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10

CLUCKERS, RAF, and IMMANUEL HALUPCZOK. "QUANTIFIER ELIMINATION IN ORDERED ABELIAN GROUPS." Confluentes Mathematici 03, no. 04 (December 2011): 587–615. http://dx.doi.org/10.1142/s1793744211000473.

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11

Foulis, David J. "Compressions on partially ordered abelian groups." Proceedings of the American Mathematical Society 132, no. 12 (July 22, 2004): 3581–87. http://dx.doi.org/10.1090/s0002-9939-04-07644-0.

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12

Pulmannová, Sylvia. "Symmetries on Partially Ordered Abelian Groups." International Journal of Theoretical Physics 45, no. 4 (March 22, 2006): 802–12. http://dx.doi.org/10.1007/s10773-006-9069-4.

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13

Ettinger, J. Mark. "Finitely presented partially ordered abelian groups." Discrete Mathematics 175, no. 1-3 (October 1997): 133–41. http://dx.doi.org/10.1016/s0012-365x(96)00146-x.

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14

Pajoohesh, Homeira. "A relationship between the category of chain MV-algebras and a subcategory of abelian groups." Mathematica Slovaca 71, no. 4 (August 1, 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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15

Ploščica, Miroslav. "Cevian properties in ideal lattices of Abelian ℓ-groups." Forum Mathematicum 33, no. 6 (October 26, 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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16

Močkoř, Jiří. "Ordered groups with greatest common divisors theory." International Journal of Mathematics and Mathematical Sciences 24, no. 7 (2000): 469–79. http://dx.doi.org/10.1155/s0161171200004087.

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An embedding (called aGCD theory) of partly ordered abelian groupGinto abelian l-groupΓis investigated such that any element ofΓis an infimum of a subset (possible non-finite) fromG. It is proved that a GCD theory need not be unique. Acomplete GCD theoryis introduced and it is proved thatGadmits a complete GCD theory if and only if it admits a GCD theoryG→Γsuch thatΓis an Archimedean l-group. Finally, it is proved that a complete GCD theory is unique (up too-isomorphisms) and that a po-group admits the complete GCD theory if and only if anyv-ideal isv-invertible.

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17

Glass, A. M. W. "Corrigendum to “Weakly Abelian lattice-ordered groups”." Proceedings of the American Mathematical Society 130, no. 3 (October 11, 2001): 925–26. http://dx.doi.org/10.1090/s0002-9939-01-06502-9.

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18

Kumar, Manish. "Valuations and rank of ordered abelian groups." Proceedings of the American Mathematical Society 133, no. 2 (August 25, 2004): 343–48. http://dx.doi.org/10.1090/s0002-9939-04-07692-0.

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19

Goncharov, Sergey S., Steffen Lempp, and Reed Solomon. "The computable dimension of ordered abelian groups." Advances in Mathematics 175, no. 1 (April 2003): 102–43. http://dx.doi.org/10.1016/s0001-8708(02)00042-7.

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20

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

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21

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

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22

Oman, Greg, Caitlin Randall, and Logan Robinson. "Infinite sums in totally ordered abelian groups." Involve, a Journal of Mathematics 12, no. 2 (January 1, 2019): 281–300. http://dx.doi.org/10.2140/involve.2019.12.281.

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23

Chajda, Ivan, Jan Paseka, and Lei Qiang. "On Realization of Partially Ordered Abelian Groups." International Journal of Theoretical Physics 52, no. 6 (December 5, 2012): 2028–37. http://dx.doi.org/10.1007/s10773-012-1426-x.

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24

Dahn, Bernd I., and Wolfgang Lenski. "Universally generic finitely generated ordered Abelian groups." Order 11, no. 1 (March 1994): 77–84. http://dx.doi.org/10.1007/bf01462231.

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25

Wehrung, Friedrich. "Monoids of Intervals of Ordered Abelian Groups." Journal of Algebra 182, no. 1 (May 1996): 287–328. http://dx.doi.org/10.1006/jabr.1996.0172.

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26

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

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27

Veisi, Amir, and Ali Delbaznasab. "Metric spaces related to Abelian groups." Applied General Topology 22, no. 1 (April 1, 2021): 169. http://dx.doi.org/10.4995/agt.2021.14446.

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When working with a metric space, we are dealing with the additive group (R, +). Replacing (R, +) with an Abelian group (G, ∗), offers a new structure of a metric space. We call it a G-metric space and the induced topology is called the G-metric topology. In this paper, we are studying G-metric spaces based on L-groups (i.e., partially ordered groups which are lattices). Some results in G-metric spaces are obtained. The G-metric topology is defined which is further studied for its topological properties. We prove that if G is a densely ordered group or an infinite cyclic group, then every G-metric space is Hausdorff. It is shown that if G is a Dedekind-complete densely ordered group, (X, d) a G-metric space, A ⊆ X and d is bounded, then f : X → G with f(x) = d(x, A) := inf{d(x, a) : a ∈ A} is continuous and further x ∈ clXA if and only if f(x) = e (the identity element in G). Moreover, we show that if G is a densely ordered group and further a closed subset of R, K(X) is the family of nonempty compact subsets of X, e < g ∈ G and d is bounded, then d′ (A, B) < g if and only if A ⊆ Nd(B, g) and B ⊆ Nd(A, g), where Nd(A, g) = {x ∈ X : d(x, A) < g}, dB(A) = sup{d(a, B) : a ∈ A} and d′ (A, B) = sup{dA(B), dB(A)}.

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28

Gluschankof, Daniel, and François Lucas. "Hyper-regular lattice-ordered groups." Journal of Symbolic Logic 58, no. 4 (December 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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29

Gozali, Sumanang Muhtar, Dian Latifah, Rizky Rosjanuardi, and Imam Nugraha Albania. "Order Ideals on Lexicographic Direct Sum of Three Totally Ordered Abelian Groups." CAUCHY: Jurnal Matematika Murni dan Aplikasi 9, no. 1 (May 16, 2024): 66–72. http://dx.doi.org/10.18860/ca.v9i1.24116.

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Order ideals play an important role in the study of abstract algebra, especially in the study of ordered groups. In this paper, we focus on the study of order ideals in lexicographic direct sums of totally ordered Abelian groups. We begin by examining the order ideals in the group of integers , and the group of real numbers It is shown that there are no non-trivial order ideals in both groups. Next, we revisit the order ideals in the lexicographic direct sum of two totally ordered Abelian groups, The only non-trivial order ideal of is Furthermore, our study extends to the lexicographic direct sum of three totally ordered Abelian groups: and We investigate the non-trivial order ideals in these structures. It is stated that the non-trivial order ideals of are only and Furthermore, the non-trivial order ideals of are only and .Keywords: order ideal; lexicographic order; direct sum.

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30

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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31

Dolich, Alfred, and John Goodrick. "A Characterization of Strongly Dependent Ordered Abelian Groups." Revista Colombiana de Matemáticas 52, no. 2 (July 1, 2018): 139–59. http://dx.doi.org/10.15446/recolma.v52n2.77154.

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We characterize all ordered Abelian groups whose ﬁrst-order theory in the language {+, <, 0} is strongly dependent. The main result of this note was obtained independently by Halevi and Hasson [7] and Farré [5].

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32

Gluschankof, Daniel. "The elementary class of products of totally ordered abelian group." Journal of Symbolic Logic 56, no. 1 (March 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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33

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 (June 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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34

Elliott, George A. "Book Review: Partially ordered abelian groups with interpolation." Bulletin of the American Mathematical Society 21, no. 1 (July 1, 1989): 200–205. http://dx.doi.org/10.1090/s0273-0979-1989-15822-9.

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35

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

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36

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

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37

Lehéricy, Gabriel. "A structure theorem for abelian quasi-ordered groups." Journal of Pure and Applied Algebra 223, no. 8 (August 2019): 3238–61. http://dx.doi.org/10.1016/j.jpaa.2018.10.016.

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38

Caillot, J. F., and F. Wehrung. "Finitely Presented, Coherent, and Ultrasimplicial Ordered Abelian Groups." SemiGroup Forum 61, no. 1 (July 2000): 116–37. http://dx.doi.org/10.1007/pl00006008.

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39

Rosenthal, Kimmo I. "Corrigendum: A completion for partially ordered abelian groups." Semigroup Forum 32, no. 1 (December 1985): 124. http://dx.doi.org/10.1007/bf02575528.

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40

Kuhlmann, Katarzyna, Franz-Viktor Kuhlmann, and Saharon Shelah. "Symmetrically complete ordered sets abelian groups and fields." Israel Journal of Mathematics 208, no. 1 (September 2015): 261–90. http://dx.doi.org/10.1007/s11856-015-1199-z.

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41

Halevi, Yatir, and Assaf Hasson. "Strongly dependent ordered abelian groups and Henselian fields." Israel Journal of Mathematics 232, no. 2 (June 20, 2019): 719–58. http://dx.doi.org/10.1007/s11856-019-1885-3.

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42

Qingxiang, Xu. "Toeplitz algebras on discrete abelian quasily ordered groups." Proceedings of the American Mathematical Society 128, no. 5 (August 5, 1999): 1405–8. http://dx.doi.org/10.1090/s0002-9939-99-05268-5.

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43

Weispfenning, V. "Existential equivalence of ordered abelian groups with parameters." Archive for Mathematical Logic 29, no. 4 (December 1990): 237–48. http://dx.doi.org/10.1007/bf01651327.

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44

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

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45

Wehrung, Friedrich. "Spectral spaces of countable Abelian lattice-ordered groups." Transactions of the American Mathematical Society 371, no. 3 (October 23, 2018): 2133–58. http://dx.doi.org/10.1090/tran/7596.

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46

Ehrlich, P. "Dedekind cuts of Archimedean complete ordered abelian groups." Algebra Universalis 37, no. 2 (April 1, 1997): 223–34. http://dx.doi.org/10.1007/s000120050014.

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47

Fornasiero, Antongiulio, and Marcello Mamino. "Arithmetic of Dedekind cuts of ordered Abelian groups." Annals of Pure and Applied Logic 156, no. 2-3 (December 2008): 210–44. http://dx.doi.org/10.1016/j.apal.2008.05.001.

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48

EHRLICH, PHILIP, and ELLIOT KAPLAN. "NUMBER SYSTEMS WITH SIMPLICITY HIERARCHIES: A GENERALIZATION OF CONWAY’S THEORY OF SURREAL NUMBERS II." Journal of Symbolic Logic 83, no. 2 (February 5, 2018): 617–33. http://dx.doi.org/10.1017/jsl.2017.9.

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AbstractIn [16], the algebraico-tree-theoretic simplicity hierarchical structure of J. H. Conway’s ordered field ${\bf{No}}$ of surreal numbers was brought to the fore and employed to provide necessary and sufficient conditions for an ordered field to be isomorphic to an initial subfield of ${\bf{No}}$, i.e., a subfield of ${\bf{No}}$ that is an initial subtree of ${\bf{No}}$. In this sequel to [16], analogous results for ordered abelian groups and ordered domains are established which in turn are employed to characterize the convex subgroups and convex subdomains of initial subfields of ${\bf{No}}$ that are themselves initial. It is further shown that an initial subdomain of ${\bf{No}}$ is discrete if and only if it is a subdomain of ${\bf{No}}$’s canonical integer part ${\bf{Oz}}$ of omnific integers. Finally, making use of class models the results of [16] are extended by showing that the theories of nontrivial divisible ordered abelian groups and real-closed ordered fields are the sole theories of nontrivial densely ordered abelian groups and ordered fields all of whose models are isomorphic to initial subgroups and initial subfields of ${\bf{No}}$.

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49

Rangan, G. "On orderability of topological groups." International Journal of Mathematics and Mathematical Sciences 8, no. 4 (1985): 747–54. http://dx.doi.org/10.1155/s0161171285000837.

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A necessary and sufficient condition for a topological group whose topology can be induced by a total order compatible with the group structure is given and such groups are called ordered or orderable topological groups. A separable totally disconnected ordered topological group is proved to be non-archimedean metrizable while the converse is shown to be false by means of an example. A necessary and sufficient condition for a no-totally disconnected locally compact abelian group to be orderable is also given.

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50

Gluschankof, D. "The Hahn representation theorem for ℓ-groups in ZFA." Journal of Symbolic Logic 65, no. 2 (June 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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Journal articles on the topic 'Ordered Abelian groups'

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