{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,21]],"date-time":"2025-10-21T15:37:17Z","timestamp":1761061037147,"version":"build-2065373602"},"reference-count":26,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2020,5,8]],"date-time":"2020-05-08T00:00:00Z","timestamp":1588896000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The continuous and injective embeddings of closed curves in Hausdorff topological spaces maintain isometry in subspaces generating components. An embedding of a circle group within a topological space creates isometric subspace with rotational symmetry. This paper introduces the generalized algebraic construction of functional groups and its topological embeddings into normal spaces maintaining homeomorphism of functional groups. The proposed algebraic construction of functional groups maintains homeomorphism to rotationally symmetric circle groups. The embeddings of functional groups are constructed in a sequence in the normal topological spaces. First, the topological decomposition and associated embeddings of a generalized group algebraic structure in the lower dimensional space is presented. It is shown that the one-point compactification property of topological space containing the decomposed group embeddings can be identified. Second, the sequential topological embeddings of functional groups are formulated. The proposed sequential embeddings follow Schoenflies property within the normal topological space. The preservation of homeomorphism between disjoint functional group embeddings under Banach-type contraction is analyzed taking into consideration that the underlying topological space is Hausdorff and the embeddings are in a monotone class. It is shown that components in a monotone class of isometry are not separable, whereas the multiple disjoint monotone class of embeddings are separable. A comparative analysis of the proposed concepts and formulations with respect to the existing structures is included in the paper.<\/jats:p>","DOI":"10.3390\/sym12050789","type":"journal-article","created":{"date-parts":[[2020,5,8]],"date-time":"2020-05-08T11:26:00Z","timestamp":1588937160000},"page":"789","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["The Sequential and Contractible Topological Embeddings of Functional Groups"],"prefix":"10.3390","volume":"12","author":[{"given":"Susmit","family":"Bagchi","sequence":"first","affiliation":[{"name":"Department of Aerospace and Software Engineering (Informatics), Gyeongsang National University, Jinju 660701, Korea"}]}],"member":"1968","published-online":{"date-parts":[[2020,5,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"353","DOI":"10.1016\/0166-8641(92)90108-C","article-title":"More topological applications of generic elementary embeddings","volume":"Volume 44","author":"Tall","year":"1992","journal-title":"Topology and Its Applications"},{"key":"ref_2","first-page":"440","article-title":"Embeddings of topological manifolds","volume":"19","author":"Pedersen","year":"1975","journal-title":"Ill. J. Math."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1016\/S0166-8641(96)00153-8","article-title":"Applications of braid group techniques to the decomposition of moduli spaces, new examples","volume":"Volume 78","author":"Robb","year":"1997","journal-title":"Topology and Its Applications"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Wolfram, T., and Ellialtioglu, S. (2014). Applications of Group Theory to Atoms, Molecules and Solids, Cambridge University Press.","DOI":"10.1017\/CBO9781139236294"},{"key":"ref_5","first-page":"204","article-title":"Nonuniqueness of semidirect decompositions for semidirect products with directly decomposable factors and applications for dihedral groups","volume":"23","author":"Daugulis","year":"2017","journal-title":"Algebra Discret. Math."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Linckelmann, M. (2018). The Block Theory of Finite Group Algebras, Cambridge University Press.","DOI":"10.1017\/9781108349307"},{"key":"ref_7","unstructured":"Loebl, E.M. (1975). Group Theory and Its Applications, Academic Press Inc."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1490","DOI":"10.1017\/S096012951300025X","article-title":"Borel and hausdorff hierarchies in topological spaces of choquet games and their effectivization","volume":"25","author":"Becher","year":"2015","journal-title":"Math. Struct. Comp. Sci."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Xiao, Z., Sanchez, I., and Tkachenko, M. (2019). Topological groups whose closed subgroups are separable, and the product operation. Topology and Its Applications, Elsevier.","DOI":"10.1016\/j.topol.2019.02.041"},{"key":"ref_10","first-page":"159","article-title":"On H-closed and minimal hausdorff spaces","volume":"60","author":"Joseph","year":"1976","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1007\/s002290170048","article-title":"A primary onstruction to topological embeddings and its applications","volume":"Volume 104","author":"Biasi","year":"2001","journal-title":"Manuscripta Mathematica"},{"key":"ref_12","first-page":"85","article-title":"The jordan and schoenflies theorems in axiomatic geometry","volume":"9","author":"Guggenheimer","year":"1978","journal-title":"Am. Math. Mon."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"321","DOI":"10.4153\/CJM-1967-023-8","article-title":"Embedding P-like compacta in manifolds","volume":"19","author":"McCord","year":"1967","journal-title":"Can. J. Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1112\/blms\/12.1.34","article-title":"A proof of the jordan curve theorem","volume":"12","author":"Tverberg","year":"1980","journal-title":"Bull. Lond. Math. Soc."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Little, C.H.C., and Vince, A. (1990). Embedding schemes and the jordan curve theorem. Topics in Combinatorics and Graph Theory, Physica-Verlag.","DOI":"10.1007\/978-3-642-46908-4_55"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"641","DOI":"10.1007\/s00407-014-0138-4","article-title":"Embedding semigroups in groups: Not as simple as it might seam","volume":"68","author":"Hollings","year":"2014","journal-title":"Arch. Hist. Exact Sci."},{"key":"ref_17","unstructured":"Petrashen, M.I., and Trifonov, E.D. (2009). Applications of Group Theory in Quantum Mechanics, Dover Publications Inc."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Bagchi, S. (2020). Projective and non-projective varieties of topological decomposition of groups with embeddings. Symmetry, 12.","DOI":"10.3390\/sym12030450"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s10231-016-0559-6","article-title":"The group generated by the round functions of a GOST-like cipher","volume":"196","author":"Aragona","year":"2017","journal-title":"Ann. Math. Pur. Appl."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"625","DOI":"10.1007\/s11587-018-0393-9","article-title":"Characterized subgroups of the circle group","volume":"67","author":"Santo","year":"2018","journal-title":"Ric. Math."},{"key":"ref_21","unstructured":"Vasco, M.I.G., and Steinwandt, R. (2015). Group Theoretic Cryptography, Chapman & Hall\/CRC."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"103486","DOI":"10.1016\/j.geomphys.2019.103486","article-title":"The groupoid structure of groupoid morphisms","volume":"145","author":"Chen","year":"2019","journal-title":"J. Geom. Phys."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"302","DOI":"10.1080\/00029890.1945.11991573","article-title":"What is a topological group?","volume":"52","author":"Montgomery","year":"1945","journal-title":"Am. Math. Mon."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"882","DOI":"10.1080\/00029890.2007.11920481","article-title":"The jordan curve theorem- formally and informally","volume":"114","author":"Hales","year":"2007","journal-title":"Am. Math. Mon."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"154","DOI":"10.1007\/978-3-642-72905-8_11","article-title":"Partitioning topological spaces","volume":"Volume 5","author":"Weiss","year":"1990","journal-title":"Mathematics of Ramsey Theory"},{"key":"ref_26","unstructured":"Bourbaki, N. (1982). A Panorama of Pure Mathematics, Academic Press."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/5\/789\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:26:43Z","timestamp":1760174803000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/5\/789"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,5,8]]},"references-count":26,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2020,5]]}},"alternative-id":["sym12050789"],"URL":"https:\/\/doi.org\/10.3390\/sym12050789","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2020,5,8]]}}}