×

On some types of kernels of a convergence \(l\)-group. (English) Zbl 0748.06006

The system \(\text{Conv} G\) of all sequential convergences in an \(\ell\)- group was introduced and investigated by the reviewer [Czech. Math. J. 39(114), No. 2, 232-238 (1989; Zbl 0681.06007)]. If \(p\) is a condition concerning \(\ell\)-groups, then the \(p\)-kernel of \(G\) is defined to be the largest element of the system of all convex \(\ell\)-subgroups of \(G\) which satisfy the condition \(p\) (if such a largest element does exist). In the present paper the author studies the existence of kernels corresponding to a series of conditions related to properties of sequences. In the reviewer’s paper quoted above it was assumed that the Urysohn’s Axiom was satisfied; the results of the present paper remain valid without this axiom (though the author does not mention this fact explicitly).

MSC:

06F15 Ordered groups
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces

Citations:

Zbl 0681.06007
PDF BibTeX XML Cite
Full Text: EuDML

References:

[1] J. Burzyk: The example of FLUSHD convergence without Y. Convergence structures. Proc. Conf. Bechyně (1984), Math. Research 24, Akademie-Verlag, Berlin, 1985.
[2] P. Conrad: Lattice ordered groups. Tulane University, 1970. · Zbl 0258.06011
[3] C. J. Everett S. Ulam: On ordered groups. Trans. Amer. Math. Soc. 57, 1945, 208-216. · Zbl 0061.03406
[4] R. Frič P. Vojtáš: Diagonal conditions in sequential convergence. Convergence structures, Proc. Conf. Bechyně (1984), Math. Research 24, Akademie-Verlag, Berlin 1985.
[5] M. Harminc: Sequential convergence on abelian lattice-ordered groups. Convergence structures 1984. Mathem. Research, Band 24, Akademie Verlag, Berlin, 153-158.
[6] M. Harminc: The cardinality of the system of all convergences on an abelian lattice ordered group. Czechoslov. Math. J. 37, 1987, 533-546. · Zbl 0645.06006
[7] M. Harminc: Sequential convergences on lattice ordered groups. Czechoslov. Math. J. · Zbl 0581.06009
[8] M. Harminc: Convergences on lattice ordered groups. Dissertation, Math. Inst. Slovak Acad. Sci., 1986. · Zbl 0581.06009
[9] J. Jakubík: Kernels of lattice ordered groups defined by properties of sequences. Čas. pěst. matem. 109, 1984, 290-298. · Zbl 0556.06007
[10] J. Jakubík: Convergences and complete distributivity of lattice ordered groups. Math. Slovaca 38, 1988, 269-272. · Zbl 0662.06005
[11] J. Jakubík: On summability in convergence \(l\)-groups. Čas. pěst. matem. 113, 1988, 286-292. · Zbl 0662.06006
[12] A. Kamiński: On characterization of topological convergence. Proc. Conf. on Convergence, Szczyrk (1979), Katowice 1980, 50-70.
[13] В. М. Копытов: Решеточно упорядоченные группы. Москва 1984. · Zbl 1170.01392
[14] P. Kratochvíl: Sequential convergences generated by coneighborhoods and an example of \(D\) not \(Y\) space. Czechoslov. Math. J.
[15] W. A. J. Luxemburg A. C. Zaanen: Riesz Spaces. Vol. I, Amsterdam 1971. · Zbl 0231.46014
[16] P. Mikusiński J. Pochcial: On Mackey convergence. Bull.Polish. Acad. Sci. Vol. 31 (1983), 151-155. · Zbl 0537.46016
[17] J. Novák: On convergence groups. Czechoslov. Math. J. 20, 1970, 357-374. · Zbl 0217.08504
[18] J. Novák L. Mišík: On \(L\)-spaces of continuous functions. Matem. fyz. sborník 1, 1951, 1-17. · Zbl 0044.11901
[19] E. Pap: Funkcionalna analiza, nizovne konvergencije, neki principi funkcionalen analize. Novi Sad 1982. · Zbl 0496.46001
[20] F. Papangelou: Order convergence and topological completion of commutative latticegroups. Math. Ann. 155, 1964, 81-107. · Zbl 0131.02601
[21] J. Pochcial: An example of FLUSHK-convergence semigroup without \(M\)-property. Proc. Conference of Convergence, Szczyrk 1979, 95-96 (1980).
[22] J. Pochcial: On functional convergences. Rend.Ist. Matem. Univ. Trieste, 17, 1985, 47-54. · Zbl 0606.54002
[23] Б. 3. Вулих: Введение в теорию полуупорядоченных пространств. Москва 1961. · Zbl 1160.68305
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.