# zbMATH — the first resource for mathematics

Sequential convergences on cyclically ordered groups. (English) Zbl 0654.06017
A cyclically ordered group $$(G,+,C)$$ is a group $$(G,+)$$ with a cyclic order $$C$$. That is, $$C$$ is a set of ordered triples of pairwise disjoint elements of $$G$$ such that
(1) if $$(a,b,c)\in C$$, then $$(b,c,a)\in C$$ and, $$\forall x,y\in C$$, $$(x+a+y,x+b+y,x+c+y)\in C,$$
(2) if $$(a,b,d)\in C$$ and $$(b,c,d)\in C$$, then $$(a,c,d)\in C$$;
(3) $$(a,b,c)\in C$$ or $$(a,c,b)\in C$$.
The multiplicative group $$K$$ of complex numbers of modulus one is a cyclically ordered group with the natural cyclic order. If $$M$$ is linearly ordered then $$K\times M$$ can be cyclically ordered. Further [S. Swierczkowski, On cyclically ordered groups, Fundam. Math. 47, 161–166 (1959; Zbl 0096.01501)], if $$(G,+,C)$$ is a cyclically ordered group, then it is isomorphic to $$K\times M$$ for some linearly ordered group $$M$$.
In the present paper, the author defines a general convergence $$\mathcal L\subseteq G^N\times G$$ for arbitrary $$(G,+,C)$$. The main theorem completely classifies the convergences. It asserts that $$\mathcal L$$ must be of one of only two types, “discrete” or “ordered”. Interestingly, if $$G$$ is infinite Archimedean or meets a certain technical requirement, then $$\mathcal L$$ may be of either type. Otherwise $$\mathcal L$$ is discrete.

##### MSC:
 06F15 Ordered groups 06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
##### Keywords:
cyclically ordered group; cyclic order; convergence
Full Text:
##### References:
  FRIČ R., KOUTNÍK V.: Recent development in sequential convergence. Convergence Structures and Applications II, Abh. Akad. Wiss. DDR, Abt. Math.-Naturwiss.-Technik, 1984, Nr. 2N, Akademie-Verlag, Berlin 1984, 37-46. · Zbl 0567.54002  FUCHS L.: Partially Ordered Algebraic Systems. Oxford 1963. · Zbl 0137.02001  HARMINC M.: Sequential convergences on abelian lattice-ordered groups. Convergence Structures 1984. Mathematical Research, Band 24, Akademie-Verlag, Berlin 1985, 153-158.  HARMINC M.: The cardinality of the system of all sequential convergences on an abelian lattice ordered group. Czech. Math. J. 37, 1987, 533-546. · Zbl 0645.06006  HARMINC M.: Convergences on Lattice Ordered Groups (Slovak). Dissertation, Math. Inst. Slovak Acad. Sci., Bratislava, 1986.  JAKUBÍK J., ČERNÁK Š.: Completion of a cyclically ordered group. Czech. Math. J. 37, 1987, 157-174. · Zbl 0624.06021  MIKUSIŃSKI P.: Problems posed at the conference. Proc. Conf. on Convergence, Szczyrk 1979, Katowice, 1980, 110-112.  NOVÁK J.: On convergence groups. Czechoslovak Math. J. 20 (1970), 357-374. · Zbl 0217.08504  PRINGEROVÁ G.: Radicals on Linearly Ordered and on Cyclically Ordered Groups (Slovak). Dissertation, Komenský University, Bratislava, 1986.  RIEGER L. S.: On ordered and cyclically ordered groups I-III (Czech). Věstník Král. České Spol. Nauk, 1946, Nr. 6, 1-31; 1947, Nr. V 1-33; 1948, Nr. V 1-26.  ŚWIERCZKOWSKI S.: On cyclically ordered groups. Fund. Math. 47 (1959), 161-166. · Zbl 0096.01501  ZABARINA A. I.: On the theory of cyclically ordered groups (Russian). Matem. zametki 31 (1982), Nr. 1, 3-12. · Zbl 0492.06014  ZABARINA A. I., PESTOV G. G.: On a theorem of Świerczkowski (Russian). Sibir. Matem. ž. 25 (1984), Nr. 4, 46-53. · Zbl 0554.06013
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.