zbMATH — the first resource for mathematics

Above and below subgroups of a lattice-ordered group. (English) Zbl 0628.06013
Authors’ summary: “In an l-group G, this paper defines an l-subgroup A to be above an l-subgroup B (or B to be below A) if for every integer n, \(a\in A\) and \(b\in B\), n(\(| a| \wedge | b|)\leq | a|\). It is shown that for every l-subgroup A, there exists an l- subgroup B maximal below A which is closed, convex, and, if the l-group G is normal-valued, unique, and that for every l-subgroup B there exists an l-subgroup A maximal above B which is saturated: if \(0=x\wedge y\) and \(x+y\in A\), then \(x\in A.\)
Given l-groups A and B, it is shown that every lattice ordering of the splitting extension \(G=A\times B\), which extends the lattice orders of A and B and makes A lie above B, is uniquely determined by a lattice homomorphism \(\pi\) from the lattice of principal convex l-subgroups of A into the cardinal summands of B. This extension is sufficiently general to encompass the cardinal sum of two l-groups, the lex extension of an l- group by an o-group, and the permutation wreath product of two l-groups.
Finally, a characterization is given of those abelian l-groups G that split off below: whenever G is a convex l-subgroup of an l-group H, H is then a splitting extension of G by A for any l-subgroup A maximal above G in H.”
Reviewer: B.F.Smarda

06F15 Ordered groups
20F60 Ordered groups (group-theoretic aspects)
Full Text: DOI
[1] Richard N. Ball, Convergence and Cauchy structures on lattice ordered groups, Trans. Amer. Math. Soc. 259 (1980), no. 2, 357 – 392. · Zbl 0441.06015
[2] Richard N. Ball, The generalized orthocompletion and strongly projectable hull of a lattice ordered group, Canad. J. Math. 34 (1982), no. 3, 621 – 661. · Zbl 0503.06016
[3] -, The structure of the \( \alpha \)-completion of a lattice ordered group, Pacific J. Math. (submitted).
[4] Richard N. Ball, Topological lattice-ordered groups, Pacific J. Math. 83 (1979), no. 1, 1 – 26. · Zbl 0434.06016
[5] Alain Bigard, Groupes archimédiens et hyper-archimédiens, Séminaire P. Dubreil, M.-L. Dubreil-Jacotin, L. Lesieur et G. Pisot: 1967/68, Algèbre et Théorie des Nombres, Secrétariat mathématique, Paris, 1969, pp. Fasc. 1, Exp. 2, 13 (French).
[6] Alain Bigard, Klaus Keimel, and Samuel Wolfenstein, Groupes et anneaux réticulés, Lecture Notes in Mathematics, Vol. 608, Springer-Verlag, Berlin-New York, 1977 (French). · Zbl 0384.06022
[7] J. W. Brewer, P. F. Conrad, and P. R. Montgomery, Lattice-ordered groups and a conjecture for adequate domains, Proc. Amer. Math. Soc. 43 (1974), 31 – 35. · Zbl 0294.06016
[8] R. D. Byrd, Lattice ordered groups, Thesis, Tulane University, 1966.
[9] P. Conrad, Lattice ordered groups, Lecture Notes, Tulane University, 1970. · Zbl 0258.06011
[10] Paul Conrad, Lex-subgroups of lattice-ordered groups, Czechoslovak Math. J. 18 (93) (1968), 86 – 103 (English, with Loose Russian summary). · Zbl 0155.05902
[11] -, The structure of an \( l\)-group that is determined by its minimal prime subgroups, Ordered Groups, Lecture Notes in Pure and Appl. Math., vol. 62, Dekker, New York, 1980. · Zbl 0449.06010
[12] Paul Conrad, John Harvey, and Charles Holland, The Hahn embedding theorem for abelian lattice-ordered groups, Trans. Amer. Math. Soc. 108 (1963), 143 – 169. · Zbl 0126.05002
[13] A. M. W. Glass, Ordered permutation groups, London Mathematical Society Lecture Note Series, vol. 55, Cambridge University Press, Cambridge-New York, 1981. · Zbl 0473.06010
[14] S. McCleary, Closed cls of a normal valued \( l\)-group...
[15] E. B. Scrimger, A large class of small varieties of lattice-ordered groups, Proc. Amer. Math. Soc. 51 (1975), 301 – 306. · Zbl 0312.06010
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.