Convergences on lattice ordered groups with a finite number of disjoint elements.

*(English)*Zbl 1141.06016Convergence of sequences in totally ordered groups can be considered equivalently either (1) in terms of the interval topology, or (2) with relation to sequences of positive elements whose infimum is \(0\). Although both of these methods can be extended to lattice-ordered groups, their relationship is less exact and there are many possible generalizations which have been investigated during the last 50 years. For example, see [B. Banaschewski, Math. Nachr. 16, 51–71 (1957; Zbl 0077.03302)].

The paper under review is the 13th in a sequence by the present author and by M. Harminc, beginning in 1984 and dealing with convergence of sequences in lattice-ordered groups in the sense of (2). A congruence of a lattice-ordered group \(G\) is a set of sequences of positive elements of \(G\) which converge to \(0\) subject to a set of conditions described in the paper. These congruences are ordered by containment, which is related in a general sense to the rate of convergence. In the case of totally ordered groups, there are at most two congruences, namely those sequences which are finally \(0\), and those which are eventually bounded above by a monotone sequence whose infimum is \(0\). The author generalizes this observation to show that in a lattice-ordered group in which every set of disjoint positive elements is finite, the set of congruences is a finite Boolean algebra; and in the abelian case, the converse is also true.

The paper under review is the 13th in a sequence by the present author and by M. Harminc, beginning in 1984 and dealing with convergence of sequences in lattice-ordered groups in the sense of (2). A congruence of a lattice-ordered group \(G\) is a set of sequences of positive elements of \(G\) which converge to \(0\) subject to a set of conditions described in the paper. These congruences are ordered by containment, which is related in a general sense to the rate of convergence. In the case of totally ordered groups, there are at most two congruences, namely those sequences which are finally \(0\), and those which are eventually bounded above by a monotone sequence whose infimum is \(0\). The author generalizes this observation to show that in a lattice-ordered group in which every set of disjoint positive elements is finite, the set of congruences is a finite Boolean algebra; and in the abelian case, the converse is also true.

Reviewer: Charles W. Holland (Bowling Green)

##### MSC:

06F15 | Ordered groups |

##### Citations:

Zbl 0077.03302
Full Text:
EuDML

**OpenURL**

##### References:

[1] | CONRAD P.: The structure of a lattice-ordered group with a finite number of disjoint elements. Michigan Math. J. 7 (1960), 171-180. · Zbl 0103.01501 |

[2] | FUCHS L.: Partially Ordered Algebraic Systems. Pergamon Press, Oxford, 1963. · Zbl 0137.02001 |

[3] | HARMINC M.: Sequential convergence on abelian lattice-ordered groups. Convergence Structures. 1984. Math. Res. 24, Akademie Verlag, Berlin, 1985, pp. 153-158. |

[4] | HARMINC M.: The cardinality of the system of all convergences on an abelian lattice ordered group. Czechoslovak Math. J. 37 (1987), 533-546. · Zbl 0645.06006 |

[5] | HARMINC M.: Sequential convergences on lattice ordered groups. Czechoslovak Math. J. 39 (1989), 232-238. · Zbl 0681.06007 |

[6] | HARMINC M.: Convergences on Lattices Ordered Groups. Dissertation, Math. Inst. Slovak Acad. Sci., 1986. |

[7] | HARMINC M.-JAKUBÍK J.: Maximal convergences and minimal proper convergences in l-groups. Czechoslovak Math. J. 39 (1989), 631-640. · Zbl 0703.06011 |

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

[9] | JAKUBÍK J.: On some types of kernels of a convergence l-group. Czechoslovak Math. J. 39 (1989), 239-247. · Zbl 0748.06006 |

[10] | JAKUBÍK J.: Lattice ordered groups having a largest convergence. Czechoslovak Math. J. 39 (1989), 717-729. · Zbl 0713.06009 |

[11] | JAKUBÍK J.: Convergences and higher degrees of distributivity of lattice ordered groups and of Boolean algebras. Czechoslovak Math. J. 40 (1990), 453-458. · Zbl 0731.06010 |

[12] | JAKUBÍK J.: Sequential convergences in l-groups without Urysohn’s axiom. Czechoslovak Math. J. 42 (1992), 101-116. · Zbl 0770.06008 |

[13] | JAKUBÍK J.: Nearly disjoint sequences in convergence l-groups. Math. Bohem. 125 (2000), 139-144. · Zbl 0967.06013 |

[14] | JAKUBÍK J.: On iterated limits of subsets of a convergence l-groups. Math. Bohem. 126 (2001), 53-61. · Zbl 0978.06008 |

[15] | JAKUBÍK J.: Konvexe Ketten in l-Gruppen. Časopis Pest. Mat. 84 (1959), 53-63. · Zbl 0083.01803 |

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.