Geodesic
Representation of Steinitz's lattice in lattices of~substructures of relational structures
Algebra and discrete mathematics, Tome 21 (2016) no. 2, pp. 184-201.

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General conditions under which certain relational structure contains a lattice of substructures isomorphic to Steinitz's lattice are formulated. Under some natural restrictions we consider relational structures with the lattice containing a sublattice isomorphic to the lattice of positive integers with respect to divisibility. We apply to this sublattice a construction that could be called “lattice completion”. This construction can be used for different types of relational structures, in particular for universal algebras, graphs, metric spaces etc. Some examples are considered.
Keywords: lattice, supernatural numbers, Boolean algebra.
Mots-clés : relational structure 
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Oksana Bezushchak; Bogdana Oliinyk; Vitaliy Sushchansky. Representation of Steinitz's lattice in lattices of~substructures of relational structures. Algebra and discrete mathematics, Tome 21 (2016) no. 2, pp. 184-201. http://geodesic.mathdoc.fr/item/ADM_2016_21_2_a2/

[1] E. Steinitz, “Algebraische Theorie der Korper”, J. reine angew. Math., 137 (1910), 167–309 ; Reprinted: Algebraische Theorie der Korper, Chelsea Publ, New York, 1950 | MR | Zbl

[2] Joel V. Brawley, George E. Schnibben, Infinite algebraic extensions of finite fields, Cotemporary Math., 95, Amer. Math. Soc., Providence, Rhode Island, 1989 | MR | Zbl

[3] J. G. Glimm, “On a certain class of operator algebras”, Trans. Amer. Math. Soc., 95:2 (1960), 318–340 | DOI | MR | Zbl

[4] S. C. Power, Limit algebras: an introduction to subalgebras of $C^{*}$-algebras, Longman Scientific and Technical, Harlow, Essex; Wiley, New York, 1993 | MR

[5] N. V. Kroshko, V. I. Sushchanskij, “Homogeneous symmetric groups”, Dopov. Akad. Nauk Ukr., 12 (1993), 9–13 (Ukrainian) | MR | Zbl

[6] N. Kroshko, V. Sushchansky, “Direct limits of symmetric and alternating groups with strictly diagonal embeddings”, Arch. Math. (Basel), 71 (1998), 173–182 | DOI | MR | Zbl

[7] E. E. Goryachko, “The $K_0$-functor and characters of the group of rational rearrangements”, J. Math. Sci., 158:6 (2009), 838–844 (English. Russian original) | DOI | MR | Zbl

[8] E. E. Goryachko, F. V. Petrov, “Indecomposable characters of the group of rational rearrangements of the segment”, J. Math. Sci., 174 (2011), 7–14 (English. Russian original) | DOI | MR | Zbl

[9] A. Bier, V. I. Sushchanskyy, “Dense subgroups in the group of interval exchange transformations”, Algebra and Discrete Math., 17:2 (2014), 232–247 | MR | Zbl

[10] V. I. Sushchanskij, “The lattice of supernatural numbers as a subalgebra lattice in universal algebras”, Dopov. Akad. Nauk Ukr., 11 (1997), 42–45 (Ukrainian)

[11] V. I. Sushchansky, V. S. Sikora, Operations on permutation groups. The theory and application, Ruta, Chernivtsi, 2003 (Ukrainian)

[12] Peter J. Cameron, Oligomorphic Permutation Groups, London Math. Soc. Lecture Notes, 152, Cambridge Univ. Press, 1990 | MR | Zbl

[13] N. Bourbaki, Set theory, Mir, Moskva, 1965 (Russian)

[14] A. E. Zalesskii, “Group rings of inductive limits of alternating groups”, Leningrad Mathematical Journal, 2:6 (1991), 1287–1303 | MR

[15] F. Blanchard, E. Formenti, P. Kurka, “Cellular Automata in the Cantor, Besicovitch and Weyl Topological Spaces”, Complex Systems, 11:2 (1997), 107–123 | MR | Zbl

[16] B. V. Oliynyk, V. I. Sushchanskii, “The isometry groups of Hamming spaces of periodic sequences”, Siberian Mathematical Journal, 54:1 (2013), 124–136 | DOI | MR | Zbl

[17] P. J. Cameron, S. Tarzi, “Limits of cubes”, Topology and its Applications, 155:14 (2008), 1454–1461 | DOI | MR | Zbl

[18] B. Oliynyk, “The diagonal limits of Hamming spaces”, Algebra and Discrete Math., 15:2 (2013), 229–236 | MR | Zbl

[19] O. O. Bezushchak, V. I. Sushchansky, “Groups of periodically defined linear transformations of an infinite-dimensional vector space”, Ukrainian Mathematical Journal, 67:10 (2016), 1457–1468 | DOI | MR