Convergences and higher degrees of distributivity of lattice ordered groups and of Boolean algebras. (English) Zbl 0731.06010

The 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.


06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
06E99 Boolean algebras (Boolean rings)
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[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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