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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.

06F15 Ordered groups
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
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