×

zbMATH — the first resource for mathematics

Radical classes of cyclically ordered groups. (English) Zbl 0662.06004
The notions of a radical class and a K-radical class of linearly ordered groups (introduced by C. G. Chehata and R. Wiegandt [Math. Rév. Anal. Numér. Théor. Approx. 20, 143-157 (1978; Zbl 0409.06008)] and by J. Jakubík [Math. Slovaca 38, 33-44 (1988; Zbl 0646.06012)] are extended to the case of abelian cyclically ordered groups. Denote by \({\mathcal R}\) the lattice of all radical classes of abelian linearly ordered groups; \({\mathcal R}_{{\mathcal K}}\) be the lattice of all K-radical classes of abelian linearly ordered groups; \({\mathcal R}_{{\mathcal C}}\) be the lattice of all radical classes of abelian cyclically ordered groups; \({\mathcal R}_{{\mathcal K}{\mathcal C}}\) be the lattice of all K-radical classes of abelian cyclically ordered groups. In the paper under review order properties of \({\mathcal R}_{{\mathcal C}}\) and \({\mathcal R}_{{\mathcal K}{\mathcal C}}\) and relations between \({\mathcal R}\), \({\mathcal R}_{{\mathcal K}}\), \({\mathcal R}_{{\mathcal C}}\) and \({\mathcal R}_{{\mathcal K}{\mathcal C}}\) are investigated. From the results: \({\mathcal R}_{{\mathcal C}}\) contains atoms and dual atoms; \({\mathcal R}_{{\mathcal K}}\) is isomorphic to \({\mathcal R}_{{\mathcal K}{\mathcal C}}\) but not a sublattice of \({\mathcal R}_{{\mathcal K}{\mathcal C}}\); \({\mathcal R}\) is a retract of \({\mathcal R}_{{\mathcal C}}\).
Reviewer: M.Harminc

MSC:
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
PDF BibTeX XML Cite
Full Text: EuDML
References:
[1] CHEHATA C. G., WIEGANDT R.: Radical theory for fully ordered groups. Mathematica - Rév. d’Anal. Numér. Théor. Approx. 20 (43), 1979, 143-157. · Zbl 0409.06008
[2] CONRAD P.: K-radical classes of lattice ordered groups. Conf. Algebra, Carbondale, 1980, Lect. Notes in Math. 848. · Zbl 0455.06010
[3] Фукс Л.: Частично упорядоченные алгебраические системы. Москва 1972. · Zbl 1225.01023
[4] GARDNER B. J.: Some aspects of radical theory for fully ordered abelian groups. Comment. math. Univ. Carol. (CMUC) 26, 1985, 821-837. · Zbl 0584.06010
[5] JAKUBÍK J.: On the lattice of radical classes of linearly ordered groups. Studia scient. mathem. Hungar. 19, 198 V 76-86.
[6] JAKUBÍK J.: On the lattice of semisimple classes of linearly ordered groups. Čas. pěst. matem. 107, 1982, 183-190.
[7] JAKUBÍK J.: On radical classes of abelian linearly ordered groups. Math. Slovaca 35, 1985, 141-154. · Zbl 0579.20034
[8] JAKUBÍK J.: K-radical classes of abelian linearly ordered groups. Math. Slovaca 38, 1988, 33-44. · Zbl 0646.06012
[9] JAKUBÍK J., ČERNÁK Š.: Completion of a cyclically ordered group. Czechosl. Math. J. 37, 1987,157-174. · Zbl 0624.06021
[10] JAKUBÍK J., PRINGEROVÁ G.: Representations of cyclically ordered groups. Čas. pěst. matem. 113, 1988, 197-208. · Zbl 0654.06016
[11] JAKUBÍKOVÁ M.: Hereditary radical classes of linearly ordered groups. Čas. pěst. matem. 107, 1982, 199-207.
[12] PRINGEROVÁ G.: Covering condition in the lattice of radical classes of linearly ordered groups. Math. Slovaca 33, 1983, 363-369. · Zbl 0519.06015
[13] PRINGEROVÁ G.: On semisimple classes of abelian linearly ordered groups. Čas. pěst. matem. 108, 1983, 40-52. · Zbl 0516.06013
[14] RIEGER L.: O uspořádaných a cyklicky uspořádaných grupách I, II, III. Věstník král. české spol. nauk 1946, 1-31; 1947, 1-33; 1948, 1-26.
[15] SWIERCZKOWSKI S.: On cyclically ordered groups. Fundament. Math. 47, 1959, 161-166. · Zbl 0096.01501
[16] ЗАБАРИНА А. И.: К теории сиклически упорядоченных групп. Матем. заметки 31, 1982, 3-12. · Zbl 1239.01036
[17] ЗАБАРИНА А. И.: О линейном и циклическом порядках в группе. Сибир. матем. ж. 26, 1985, 204-207. · Zbl 1239.11074
[18] ЗАБАРИНА А. И., ПЕСТОВ Г. Г.: К теореме Сврчковского. Сибир. матем. ж. 25, 1984, 46-53. · Zbl 1239.01039
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.