×

zbMATH — the first resource for mathematics

On convex linearly ordered subgroups of an \(h\ell \)-group. (English) Zbl 0986.06011
Let \(G\) be a right partially ordered group. Let us denote by \(G\uparrow \) (and \(G\downarrow \)) the set of all \(x\in G\) such that for all \(y,z\in G\) the implication \(y\leq z \Rightarrow xy\leq xz\) (and \(xy\geq xz\), respectively) is valid. Then \(G\) is called a half lattice-ordered group (an \(h\ell \)-group) if the partial order on \(G\) is not trivial, \(G={G\uparrow}\cup {G\downarrow}\) , and \({G\uparrow}\) is a lattice. Then \(G\uparrow \) is a lattice-ordered group (an \(\ell \)-group) and a normal subgroup of \(G\). Let \(\mathcal H_1\) be the class of all \(h\ell\)-groups which are not \(\ell \)-groups. The notion of an \(h\ell\)-group was introduced by M. Giraudet and F. Lucas [Fundam. Math. 139, 75-89 (1991; Zbl 0766.06014)], where it is also proved that if \(G\in \mathcal H_1\), and if the \(\ell \)-group \(G\uparrow \) is linearly ordered, then \(G\uparrow \) is abelian. Moreover, \(G\uparrow \) is in this case a maximal convex linearly ordered subgroup of \(G\). The authors of the paper, among others, generalize this result proving the following theorem: Let \(G\in \mathcal H_1\) and let \(X\) be a maximal convex linearly ordered subgroup of \(G\). If there exists \(a\in G\) such that \(e\neq a\), \(a^2=e\) and \(aX=Xa\), then \(X\) is an abelian \(\ell \)-group.

MSC:
06F15 Ordered groups
PDF BibTeX XML Cite
References:
[1] ČERNÁK S.: On the maximal Dedekind completion of a half partially ordered group. Math. Slovaca 47 (1996), 379-390. · Zbl 0888.06009
[2] ČERNÁK S.: Cantor extension of a half lattice ordered group. Math. Slovaca 48 (1998), 221-231. · Zbl 0938.06014
[3] GIRAUDET M.-LUCAS F.: Groupes á moitié ordonnés. Fund. Math. 139 (1991), 75-89. · Zbl 0766.06014
[4] GIRAUDET M.-RACHŮNEK J.: Varieties of half lattice-ordered groups of monotonic permutations in chains. Prepublication No 57, Universite Paris 7, CNRS Logique, 1996.
[5] JAKUBÍK J.: On half lattice ordered groups. Czechoslovak Math. J. 46 (1996), 745-767. · Zbl 0879.06011
[6] JAKUBÍK J.: Lexicographic products of half linearly ordered groups. Czechoslovak Math. J. · Zbl 1079.06504
[7] TON, DAO-RONG: Torsion classes and torsion prime selectors of \(h\ell\)-groups. Math. Slovaca 50 (2000), 31-40. · Zbl 0955.06010
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.