Distances on free semigroups and their applications. (English) Zbl 1406.20054
Summary: In this article it is proved that for any quasimetric \(d\) on a set \(X\) with a base-point \(p_X\) there exists a maximal invariant extension \(\hat{\rho}\) on the free monoid \(F^a(X, \mathcal V)\) in a non-Burnside quasi-variety \( \mathcal V\) of topological monoids (Theorem 6.1). This fact permits to prove that for any non-Burnside quasi-variety \( \mathcal V\) of topological monoids and any \(T_0\)-space \(X\) the free topological monoid \(F(X, \mathcal V)\) exists and is abstract free (Theorem 8.1). Corollary 10.2 affirms that \(F(X, \mathcal V)\), where \(\mathcal V\) is a non-trivial complete non-Burnside quasi-variety of topological monoids, is a topological digital space if and only if \(X\) is a topological digital space.
MSC:
| 20M05 | Free semigroups, generators and relations, word problems |
| 20M07 | Varieties and pseudovarieties of semigroups |
| 22A15 | Structure of topological semigroups |
| 54E35 | Metric spaces, metrizability |
| 20F10 | Word problems, other decision problems, connections with logic and automata (group-theoretic aspects) |
References:
| [1] | Arenas F. G. Alexandroff spaces, Acta Math. Univ. Comenianae, 1999, 68, No. 1, 17–25. · Zbl 0944.54018 |
| [2] | Alexandroff P. Diskrete Raume, Mat. Sb., 1937, 2, 501–518 (in German). |
| [3] | Alexandroff P., Urysohn P. Une condition n´ec´esare et suffisante pour qu’une classe (L) soit une classe (D), C. R. Acad. Paris, 1923, 177, 1274–1276. · JFM 50.0696.01 |
| [4] | Alexandroff P., Urysohn P. M´emoire sur les espaces topologiques compacts, Verh. Akad. Wetensch. Afd. Naturk. Sect. I, Amsterdam, 1929, 14, 1–96. · JFM 55.0960.02 |
| [5] | Arhangel’skii A. V. Mappings and spaces, Uspehi Matem. Nauk, 1966, 21, vyp. 4, 133–184 (English translation: Russian Math. Surveys, 1966, 21, No. 4, 115–162). · Zbl 0171.43603 |
| [6] | Arhangel’skii A. V., Tkachenko M. G. Topological groups and related structures, Atlantis Press: Amsterdam-Paris, 2008. · Zbl 1323.22001 |
| [7] | Berinde V., Choban M. M. Generalized distances and their associate metrics. Impact on fixed point theory, Creat. Math. Inform., 2013, 22, No. 1, 23–32. · Zbl 1289.54115 |
| [8] | Birkhoff G. A Note on Topological Groups, Compositio mathematica, 1936, 3, 427–430. · Zbl 0015.00702 |
| [9] | Chittenden E. W. On the equivalence of ´ecart and voisinage, Trans. Amer. Math. Soc., 1917, 18, 161–166. · JFM 46.0288.03 |
| [10] | Choban M. M. On the theory of topological algebraic systems, Trans. Moscow Math. Soc., 1986, 48, 115–159 (Russian original: Trudy Mosk. Matem. Ob-va, 1985, 48, 106–149). · Zbl 0606.22003 |
| [11] | Choban M. M. The theory of stable metrics, Math. Balkanica, 1988, 2, 357–373. · Zbl 0681.08001 |
| [12] | Choban M. M. General conditions of the existence of free algebras, Acta Comment. Univ. Tartuensis, 1989, 836, 157–171. |
| [13] | Choban M. M. Some topics in topological algebra, Topol. Appl., 1993, 54, 183–202. · Zbl 0817.22005 |
| [14] | Choban M. M., Budanaev I. A. Distances on Monoids of Strings and Their Applications, Proceedings of the Conference on Mathematical Foundations of Informatics MFOI2016, July 25-29, 2016, Chisinau, Republic of Moldova, 2016, 144–159. · Zbl 1390.68384 |
| [15] | Choban M. M., Budanaev I. A. About Applications of Distances on Monoids of Strings, Computer Science Journal of Moldova, 2016, 24, No. 3(72), 335–356. · Zbl 1390.68384 |
| [16] | Choban M. M., Chiriac L. L. Selected problems and results of topological algebra, ROMAI J., 2013, 9, No. 1, 1–25. · Zbl 1299.08002 |
| [17] | Choban M. M., Chiriac L. L. On free groups in classes of groups with topologies, Bul. Acad. S¸tiint¸e Repub. Moldova, Mat., 2013, No. 2(72)–3(73), 61–79. · Zbl 1300.22003 |
| [18] | Dumitrascu S. S., Choban M. M. On free topological algebras with a continuous signature, Algebraic and topological systems. Matem. Issledovania, 1982, 65, 27–54. · Zbl 0503.08009 |
| [19] | Engelking R. General Topology, PWN, Warszawa, 1977. |
| [20] | Frink A. H. Distance functions and the metrization problem, Bull. Amer. Math. Soc., 1937, 43, 133–142. 118M. M. CHOBAN, I. A. BUDANAEV · JFM 63.0571.03 |
| [21] | Graev M. I. Free topological groups, Trans. Moscow Math. Soc., 1962, 8, 303–364 (Russian original: Izvestia Akad. Nauk SSSR, 1948, 12, 279–323). |
| [22] | Graev M. I. Theory of topological groups I. Norms and metrics on groups. Complete groups. Free topological groups, Uspehi Matem. Nauk, 1950, 5, No. 2(36), 3–56 (Russian). · Zbl 0037.01301 |
| [23] | Herman G. T. Geometry of Digital Spaces, Birkh¨auser, 1998. · Zbl 0902.65102 |
| [24] | Hofmann K. H., Lawson J. D., Pym J. S. The analytical and topological theory of semigroups, de Gruyter, 1990. · Zbl 0702.00013 |
| [25] | James L. M. Multiplications on spheres, I, Proceed. Amer. Mat. Soc., 1957, 13, 192–196. · Zbl 0078.37101 |
| [26] | James L. M. Multiplications on spheres, II, Trans. Amer. Math. Soc., 1957, 84, 545–548. · Zbl 0085.17202 |
| [27] | Kakutani Sh. Free topological groups and infinite direct product topological groups, Proceed. Imp. Acad. Tokyo, 1944, 20, 595–598. · Zbl 0063.03105 |
| [28] | Kakutani Sh.Uber die Metrisation der topologischen Gruppen, Proc. Imp. Acad. Japan,¨ 1936, 12, 82–84. · JFM 62.1230.03 |
| [29] | Kakutani Sh. Free topological groups and infinite direct product topological groups, Proc. Imp. Acad. Tokyo, 1944, 20, 595–598. · Zbl 0063.03105 |
| [30] | Khalimsky E. Topological structures in computer science, Journal of Applied Math. and Simulation, 1987, 1, No. 1, 25–40. · Zbl 0655.68021 |
| [31] | Khalimsky E., Kopperman R., Meyer P. R. Computer graphics and connected topologies on finite ordered sets, Topol. Appl., 1990, 36, 1–17. · Zbl 0709.54017 |
| [32] | Levenshtein V. I. Binary codes capable of correcting deletions, insertions, and reversals, DAN SSSR, 1965, 163, No. 4, 845–848 (in Russian) (English translation: Soviet Physics Doklady, 1965, 10, No. 8, 707–710). · Zbl 0149.15905 |
| [33] | Malcev A. I. Free topological algebras, Trans. Moscow Math. Soc., (2), 1961, 17, 173–200. (Russian original: Izvestia Akad. Nauk SSSR, 1957, 21, 171–198). · Zbl 0128.24702 |
| [34] | Mostert P. S. The structure of topological semigroups – revisited, Bull. Amer. Math. Soc., 1966, 72, No. 4, 601–618. · Zbl 0146.03202 |
| [35] | Mukherjea A., Tserpes N. Measures on topological semigroups, Lect. notes in math. 547, Springer, 1976. · Zbl 0342.43001 |
| [36] | Nedev S. I. o-metrizable spaces, Trudy Moskov. Mat. Ob-va, 1971, 24, 201–236 (English translation: Trans. Moscow Math. Soc., 1974, 24, 213–247). · Zbl 0295.54039 |
| [37] | Nedev S. I., Choban M. M. On the theory of o-metrizable spaces, I. Vestnik Moskovskogo Universiteta, 1972, No. 1, 8–15. (English translation: Moscow University Mathematics Bulletin, 1973, 27, No. 1–2, 5–9). · Zbl 0242.54029 |
| [38] | Nedev S. I., Choban M. M. On the theory of o-metrizable spaces, II. Vestnik Moskovskogo Universiteta, 1972, No. 2, 10–17. (English translation: Moscow University Mathematics Bulletin, 1973, 27, No. 1–2, 65–70). · Zbl 0242.54030 |
| [39] | Nedev S. I., Choban M. M. On the theory of o-metrizable spaces, III. Vestnik Moskovskogo Universiteta, 1972, No. 3, 10–15. (English translation: Moscow University Mathematics Bulletin, 1973, 27, No. 3–4, 7–11). · Zbl 0248.54034 |
| [40] | Niemytzki V.On the third axiom of metric spaces, Trans Amer. Math. Soc., 1927, 29, 507–513. · JFM 53.0558.01 |
| [41] | Niemytzki V. Uber die Axiome des metrischen Raumes, Math. Ann., 1931, 104, 666–671. · JFM 57.0734.02 |
| [42] | Romaguera S., Sanchis M., Tkachenko M. Free paratopological groups, Topology Proceed., 2003, 27, No. 2, 613–640. DISTANCES ON FREE SEMIGROUPS AND THEIR APPLICATIONS119 · Zbl 1062.22005 |
| [43] | Ruppert W. A. F. Compact semitopological semigroups: an intrinsic theory, Lect. notes in math. 1079, Springer, 1984. · Zbl 0606.22001 |
| [44] | Swierczkowski S. Topologies in free algebras, Proceed. London Math. Soc., 1964, 15 (55), 566–576. · Zbl 0123.09902 |
| [45] | Wallace A. D. The structure of topological semigroups, Bull. Amer. Math. Soc., 1955, 61, No. 2, 95–112. · Zbl 0065.00802 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
