Abstract
In this paper, we discuss the notions of minimal ideals, maximal ideals and principal ideals on a topological ternary semigroup. We express the kernel of a topological ternary semigroup in different ways and provide characterizations of kernel in topological ternary semigroups. We establish equivalences between minimal left ideal, minimal ideal and maximal ternary subgroup on a paratopological ternary group.
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Carruth, J.H., Hildebrant, J.A., Koch, R.J.: The Theory of Topological Semigroups, vol. 1. Marcel Dekker, Inc., New York (1983)
Choosuwan, P., Chinram, R.: A study on quasi-ideals in ternary semigroups. Int. J. Pure Appl. Math. 77(5), 639–647 (2012)
Dixit, N.V., Dewan, S.: A note on quasi and bi-ideals in ternary semigroups. Int. J. Math. Math. Sci. 18, 501–508 (1995)
Dörnte, W.: Untersuchungen über einen verallgemeinerten Gruppenbegriff. Math. Z. 29, 1–19 (1928)
Dudek, W. A., Grozdzinska, I.: On ideals in regular \(n\)-semigroups. Matematicki Bilten (Skopje), 3/4(XXIX/XXX), 35–44 (1979–1980)
Dutta, T.K., Kar, S., Maity, B.K.: On ideals in regular ternary semigroups. Discuss Math. General Algebra Appl. 28, 147–159 (2008)
Iampan, A.: Some properties of ideal extension in ternary semigroups. Iran. J. Math. Sci. Inf. 8(1), 67–74 (2013)
Kar, S., Maity, B.K.: Congruences on ternary semigroups. J. Chungcheong Math. Soc. 20(3), 191–201 (2007)
Kar, S., Maity, B.K.: Some ideals of ternary semigroups. Analele ŞTiinŢifice ale universitĂŢII “AL.I. CUZA” DIN IAŞI (S.N.) MatematicĂ, Tomul LVII, f.2,: 247–258 (2011). https://doi.org/10.2478/v10157-011-0024-1
Kasner, E.: An extension of the group concept. Bull. Am. Math. Soc. 10, 290–291 (1904)
Lehmer, D.H.: A Ternary analogue of Abelian groups. Am. J. Math. 59, 329–338 (1932)
Loś, J.: On the extending of models I. Fundam. Math. 42, 38–54 (1955)
Petchkaew, P., Chinram, R.: The minimality and Maximality of \(n\)-ideals in \(n\)-ary semigroups. Eur. J. Pure Appl. Math. 11(3), 762–773 (2018)
Post, E.L.: Polyadic groups. Trans. Am. Math. Soc. 48, 208–350 (1940)
Rao, S. G., Rao, M. D., Sivaprasad, P., Rao, M.C.H.: Maximal ideal of compact connected topological ternary semigroups. Math. Sci. Int. Res. J. 4(2), 144–148 (2015)
Sabir, M., Bano, M.: Prime bi-ideals in ternart semigroups. Quasigroups Relat. Syst. 16, 239–256 (2008)
Samanta, S., Jana, S., Kar, S.: Note on topological ternary semigroup. Asia–Eur. J. Math. 14(5), 11 pages (2021)
Sheeja, G., Sri Bala, S.: Simple ternary semigroup. Quasigroups Relat. Syst.21, 103–116 (2013)
Sioson, F.M.: Ideal theory in ternary semigroups. Math. Japonica 10, 63–84 (1965)
Samanta, S.: Rees matrix on topological ternary semigroups. Bull. Cal. Math. Soc. 113(5), 409–420 (2021)
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We are thankful to the learned referees for their valuable suggestions. We have incorporated all of their suggestions in this revised version.
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Samanta, S., Jana, S. & Kar, S. Ideals in topological ternary semigroups. Bol. Soc. Mat. Mex. 28, 21 (2022). https://doi.org/10.1007/s40590-022-00416-9
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DOI: https://doi.org/10.1007/s40590-022-00416-9
Keywords
- Ternary semigroup
- Ternary group
- Ternary subgroup
- Topological ternary semigroup
- Paratopological ternary group
- Ideal
- Kernel