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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
##### 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
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