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Completion of a cyclically ordered group. (English) Zbl 0624.06021
A cyclically ordered set is a set G with a ternary relation C which satisfies
(x,y,z)$$\in C\Rightarrow (z,y,x)\not\in C$$ (asymmetry),
(x,y,z)$$\in C\Rightarrow (y,z,x)\in C$$ (cyclicity),
(x,y,z)$$\in C$$, (x,z,u)$$\in C\Rightarrow (x,y,u)\in C$$ (transitivity),
x,y,z$$\in G$$, $$x\neq y\neq z\neq x\Rightarrow either$$ (x,y,z)$$\in C$$ or (z,y,x)$$\in C$$ (linearity).
A cyclically ordered group is a group $$(G,+)$$ such that G is a cyclically ordered set and it holds $$(x,y,z)\in C\Rightarrow (a+x+b,a+y+b,a+z+b)\in C$$ for any a,b$$\in G$$. A cut on a cyclically ordered set G is a linear order $$<$$ on G such that $$x<y<z\Rightarrow (x,y,z)\in C$$. Such a cut is called regular if $$(G,<)$$ either contains a least element or has neither a least nor a greatest element. The set C(G) of all regular cuts on G with naturally defined cyclic order is called a completion of G.
Let $$(G,+)$$, $$(G_ 1,+_ 1)$$ be cyclically ordered groups such that $$G_ 1\subseteq C(G)$$ with the induced cyclic order and $$(G,+)$$ is a subgroup of $$(G_ 1,+_ 1)$$. Then $$(G_ 1,+_ 1)$$ is called an extension of $$(G,+)$$. The set of all extensions of $$(G,+)$$ is (partially) ordered by set inclusion; its greatest element is called a completion of $$(G,+)$$. The authors give a (challenging) construction of a completion of a cyclically ordered group. Also, they derive necessary and sufficient conditions under which a given cut belongs to a completion of a cyclically ordered group $$(G,+)$$, and a necessary and sufficient condition for a cyclically group $$(G,+)$$ to be complete i.e. equal to its completion.
Reviewer: V.Novák

##### MSC:
 06F15 Ordered groups 20F60 Ordered groups (group-theoretic aspects) 06B23 Complete lattices, completions
##### References:
 [1] E. Čech: Bodové množiny. Praha 1936. [2] S. Černák: On the maximal Dedekind completion of a lattice ordered group. Math. Slovaca 29, 1979, 305-313. [3] Л. Фукс: Частично упорядоченные алгебраические системы. Москва 1965. · Zbl 1099.01519 [4] J. Jakubík: Archimedean kernel of a lattice ordered group. Czech. Math. J. 28, 1978, 140-154. · Zbl 0384.06021 [5] J. Jakubík: Maximal Dedekind completion of an abelian lattice ordered group. Czech. Math. J. 28, 1978, 611-631. · Zbl 0432.06012 · eudml:13091 [6] V. Novák: Cyclically ordered sets. (Czech.) Dissertation (DrSc), Univ. J. E. Purkyně, Brno 1984. · Zbl 0567.06002 [7] V. Novák: Cuts in cyclically ordered sets. Czech. Math. J. 34, 1984, 322-333. · Zbl 0551.06002 · eudml:13455 [8] V. Novák: Cyclically ordered sets. Czech. Math. J. 32 (1982), 460-473. · Zbl 0515.06003 [9] V. Novák M. Novotný: Dimension theory for cyclically and cocyclically ordered sets. Czech. Math. J. 33 (1983), 647-653. · Zbl 0538.06002 · eudml:13420 [10] B. C. Olticar: Right cyclically ordered groups. Canad. Math. Bull. 23 (1980), 67-70. · Zbl 0439.06012 · doi:10.4153/CMB-1980-009-3 [11] G. Pringerová: Radical classes of linearly ordered groups and cyclically ordered groups. (Slovak.) Dissertation, Komensky Univ., Bratislava 1986. [12] L. Rieger: O uspořádaných a cyklicky uspořádaných grupách I-III. Věstník král. české spol. nauk 1946, 1-31; 1947, 1-33; 1948, 1-26. [13] S. Swierczkowski: On cyclically ordered groups. Fund. Math. 47 (1959), 161-166. · Zbl 0096.01501 · eudml:213528 [14] А. И. Забарина: К теории циклически упорядоченных групп. Матем. заметки 31, 1982, 3-12. · Zbl 1171.03330 [15] А. И. Забарина Г. Г. Пестов: К теореме Сверчковского. Сибир. матем. журн. 25, 1984, 46-53. · Zbl 1228.82001 [16] С. Д. Желева: О циклически упорядоченных группах. Сибир. матем. журн. 17, 1976, 1046-1051. · Zbl 1170.01332 [17] С. Д. Желева: О полу однородно цилически упорядоченных группах. Годиш. ВУЗ, прил. матем. 17, 1981, 123-136. · Zbl 1170.01413 [18] С. Д. Желева: Циклически и Г-родно упорядоченные группы. Годиш. ВУЗ, прил. матем. 17, 1981, 137-149. · Zbl 1170.01413
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