## Above and below subgroups of a lattice-ordered group

HTML articles powered by AMS MathViewer

- by Richard N. Ball, Paul Conrad and Michael Darnel PDF
- Trans. Amer. Math. Soc.
**297**(1986), 1-40 Request permission

## Abstract:

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 $b \in B$, $n(|a| \wedge |b|) \leqslant |a|$. 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 $H$.

## References

- Richard N. Ball,
*Convergence and Cauchy structures on lattice ordered groups*, Trans. Amer. Math. Soc.**259**(1980), no. 2, 357–392. MR**567085**, DOI 10.1090/S0002-9947-1980-0567085-5 - Richard N. Ball,
*The generalized orthocompletion and strongly projectable hull of a lattice ordered group*, Canadian J. Math.**34**(1982), no. 3, 621–661. MR**663307**, DOI 10.4153/CJM-1982-042-5
—, - Richard N. Ball,
*Topological lattice-ordered groups*, Pacific J. Math.**83**(1979), no. 1, 1–26. MR**555035** - 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). MR**0250950** - Alain Bigard, Klaus Keimel, and Samuel Wolfenstein,
*Groupes et anneaux réticulés*, Lecture Notes in Mathematics, Vol. 608, Springer-Verlag, Berlin-New York, 1977 (French). MR**0552653** - 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. MR**332616**, DOI 10.1090/S0002-9939-1974-0332616-X
R. D. Byrd, - Paul Conrad,
*Lex-subgroups of lattice-ordered groups*, Czechoslovak Math. J.**18(93)**(1968), 86–103 (English, with Russian summary). MR**225697**
—, - Paul Conrad, John Harvey, and Charles Holland,
*The Hahn embedding theorem for abelian lattice-ordered groups*, Trans. Amer. Math. Soc.**108**(1963), 143–169. MR**151534**, DOI 10.1090/S0002-9947-1963-0151534-0 - A. M. W. Glass,
*Ordered permutation groups*, London Mathematical Society Lecture Note Series, vol. 55, Cambridge University Press, Cambridge-New York, 1981. MR**645351**
S. McCleary, - E. B. Scrimger,
*A large class of small varieties of lattice-ordered groups*, Proc. Amer. Math. Soc.**51**(1975), 301–306. MR**384644**, DOI 10.1090/S0002-9939-1975-0384644-7

*The structure of the*$\alpha$-

*completion of a lattice ordered group*, Pacific J. Math. (submitted).

*Lattice ordered groups*, Thesis, Tulane University, 1966. P. Conrad,

*Lattice ordered groups*, Lecture Notes, Tulane University, 1970.

*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.

*Closed cls of a normal valued*$l$-

*group*...

## Additional Information

- © Copyright 1986 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**297**(1986), 1-40 - MSC: Primary 06F15; Secondary 20E22
- DOI: https://doi.org/10.1090/S0002-9947-1986-0849464-7
- MathSciNet review: 849464