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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:
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