# zbMATH — the first resource for mathematics

Semilinearly and semilattice right ordered groups. (English) Zbl 0955.06009
A right $$\vee$$-group is a right partially ordered group $$(G,\cdot , \leq , e)$$ such that $$(G,\leq)$$ is an upper semilattice. $$G$$ is semilinear if it is updirected and for all $$a, x, y \in G$$, $$a\leq x$$ and $$a\leq y$$ imply $$x\leq y$$ or $$y\leq x$$. In this paper it is proved that any semilinear right $$\vee$$-group $$G$$ is isolated, i.e., for all $$n\in \mathbb N$$ and $$a\in G$$, $$a^n\geq e$$ implies $$a\geq e$$. It is also proved that any convex $$\vee$$-subgroup of a semilinear $$\vee$$-group is 2-isolated.
##### MSC:
 06F15 Ordered groups
##### References:
 [1] ADELEKE S. A.-DUMMETT M. A. E.-NEUMANN P. M.: On a question of Frege’s about right-ordered groups. Bull. London Math. Soc. 19 (1987), 513-521. · Zbl 0632.06022 [2] KOKORIN A. L.-KOPYTOV V. M.: Linearly Ordered Groups. Nauka, Moscow, 1972. [3] KOPYTOV V. M.-MEDVEDEV N. YA.: Right-Ordered Groups. Nauchnaya kniga, Novosibirsk, 1996 [ · Zbl 0896.06017 [4] MURA R. B.-RHEMTULLA A. H.: Orderable Groups. Marcel Dekker, New York, 1977. · Zbl 0452.06011 [5] RACHŮNEK J.: Convex directed subgroups of right ordered tree groups. Czechoslovak Math. J. 41 (116) (1991), 99-103. · Zbl 0803.06020 [6] VARAKSIN S. V.: Semilinear orders on solvable and nilpotent groups. Algebra i Logika 29 (1990), 631-636. · Zbl 0797.20028
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.