Convergences and higher degrees of distributivity of lattice ordered groups and of Boolean algebras.

*(English)*Zbl 0731.06010The partially ordered set of all sequential convergences of an Abelian lattice ordered group G is denoted by Conv G; similarly, Conv B has an analogous meaning for a Boolean algebra B. In the present paper the author proves that if G is an \((\aleph_ 0,2)\)-distributive Abelian lattice ordered group, then Conv G is a complete lattice, and, analogously, if B is an \((\aleph_ 0,2)\)-distributive Boolean algebra, then Conv B is a complete lattice. This sharpens previous results of the author [Math. Slovaca 38, 269-272 (1988; Zbl 0662.06005); Czech. Math. J. 38(113), 520-530 (1988; Zbl 0668.54002)], where complete distributivity instead of \((\aleph_ 0,2)\)-distributivity was assumed.

Some other results concerning bounded convergences in a lattice ordered group G (which were defined and searched for by the author in Czech. Math. J. 39(114), 717-729 (1989; Zbl 0713.06009)) are contained. It is shown that if e is a singular element of G (i.e., the interval [0,e] of G is a Boolean algebra) and at the same time e is a strong unit in G, then the partially ordered sets of all bounded convergences in G and of all convergences in the Boolean algebra [0,e] are isomorphic.

Some other results concerning bounded convergences in a lattice ordered group G (which were defined and searched for by the author in Czech. Math. J. 39(114), 717-729 (1989; Zbl 0713.06009)) are contained. It is shown that if e is a singular element of G (i.e., the interval [0,e] of G is a Boolean algebra) and at the same time e is a strong unit in G, then the partially ordered sets of all bounded convergences in G and of all convergences in the Boolean algebra [0,e] are isomorphic.

Reviewer: M.Harminc (Košice)

##### MSC:

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

06E99 | Boolean algebras (Boolean rings) |

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\textit{J. Jakubík}, Czech. Math. J. 40(115), No. 3, 453--458 (1990; Zbl 0731.06010)

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##### References:

[1] | P. Conrad: Lattice ordered groups. Tulane University 1970. · Zbl 0258.06011 |

[2] | M. Harminc: Sequential convergences on abelian lattice-ordered groups. Convergence structures 1984. Mathematical Research, Band 24, Akdemie Verlag, Berlin; 1985, 153-158. |

[3] | M. Harminc: The cardinality of the system of all sequential convergences on an abelian lattice ordered group. Czech. Math. J. 37, 1987, 533 - 546. · Zbl 0645.06006 |

[4] | M.Harminc: Sequential convergences on lattice ordered groups. Czech. Math. J. 39, 1989, 232-238. · Zbl 0681.06007 |

[5] | J. Jakubík: Radical mappings and radical classes of lattice ordered groups. Symposia Mathematica, Vol. 21, Academic Press, London and New York, 1977, 451 - 477. |

[6] | J. Jakubík: Convergences and complete distributivity of lattice ordered groups. Math. Slovaca 38, 1988, 269-272. · Zbl 0662.06005 |

[7] | J.Jakubík: Lattice ordered groups having a largest convergence. Czech. Math. J. 39, 1989, 717-729. · Zbl 0713.06009 |

[8] | J. Jakubík: Sequential convergences in Boolean algebras. Czech. Math. J. 38, 1988, 520-530. · Zbl 0668.54002 |

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