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Direct limits of cyclically ordered groups. (English) Zbl 0821.06015
The collection of all nonempty classes of cyclically ordered groups which are closed with respect to direct limits will be denoted by \({\mathcal C}\). The collection \({\mathcal C}\) is partially ordered by inclusion. The greatest element of \({\mathcal C}\) is the class \(C_ m\) of all cyclically ordered groups. The class of all one-element cyclically ordered groups will be denoted by \(C_ 0\). This is the least element of \({\mathcal C}\). The elements of \({\mathcal C}\) will be called direct limit classes. For \(H\in C_ m\), let \(P(H)\) be the element of \({\mathcal C}\) generated by \(H\). Further, let \(G_ 1\) be the largest linearly ordered convex subgroup of \(G\) (\(G\) a cyclically ordered group), and \(C^ 1= \{G\in C_ m\): \(G_ 1= G\}\), \(C^ 0= \{G\in C_ m\): \(G_ 1= \{0\}\}\), respectively. Then \(C^ 1\) is the class of all linearly ordered groups and clearly \(C^ 1\cap C^ 0= C_ 0\). The authors prove that \(C^ 1, C_ 0\in {\mathcal C}\). An important example of cyclically ordered groups is the set \(K\) of reals \(x\) with \(0\leq x<1\); the group operation is the addition mod 1, and for \(x,y,z\in K\), \([x,y, z]= x<y<z\) or \(y<z<x\) or \(z<x<y\). If \(\varphi\) is a homomorphism of a subgroup \(G\) of \(K\) into \(K\), \(\varphi(G)\neq \{0\}\), then \(\varphi(x) =x\) for each \(x\in G\) (Theorem 3.1). Let \(G\in C^ 0\), \(G\neq \{0\}\), then \(A= \{G'\in C_ m\): \(G'= \{0\}\) or \(G'\) is isomorphic to \(G\}\) is an atom in \({\mathcal C}\) (Theorem 4.2). Let \({\mathcal S}\) be the collection of all nonempty systems of subgroups of \(K\) containing the one-element group \(\{0\}\). Let \({\mathcal S}_ 0\) be the collection of all \({\mathcal A}\in {\mathcal S}\) satisfying the following condition: If \({\mathcal A}_ 1\) is a nonempty subsystem of \({\mathcal A}\) such that \({\mathcal A}_ 1\) is directed, then \(\bigcup {\mathcal A}_ 1\) belongs to \({\mathcal A}\). Thus we have: The interval \([C_ 0, C^ 0]\) of \({\mathcal C}\) is isomorphic to \({\mathcal S}_ 0\), fails to be a proper class and is atomic. Denote \(C^{01}= C_ 0\cup (C_ m\smallsetminus C^ 1)\). Then \(C^{01}\in {\mathcal C}\) (Theorem 5.2). For \(X\in {\mathcal C}\), put \(X_ 1= X\cap C^{01}\), \(X_ 2= X\cap C^ 1\) and \(f(X)= (X_ 1, X_ 2)\). Then the mapping \(f: {\mathcal C}\to [C_ 0, C^{01}] \times [C_ 0, C^ 1]\) is an isomorphism (onto). For \(G\in C^ 1\), we put \(\psi (G)=n\) if (i) there exist elements \(0< a_ i\in G\) \((i=1,2, \dots,n)\) such that \(a_ 1\ll a_ 2\ll \dots \ll a_ n\), (ii) if \(0< b_ j\in G\) \((j=1,2, \dots,m)\) and \(b_ 1\ll b_ 2\ll \dots \ll b_ m\), then \(m\leq n\). We set \(\psi(\{ 0\})=0\) and \(\psi(G)= \infty\) if \(\psi(G)\neq n\) for \(n=1,2,\dots\;\). let \(X\) be the class of all linearly ordered groups \(G\) such that \(\psi(G) \leq n\). Then \(X\in {\mathcal C}\), \(X\leq C^ 1\) and there is no atom \(A\) in \({\mathcal C}\) with \(A\leq X\). The “natural” candidates of being atoms of \({\mathcal C}\) seem to be the limit classes \(P(Z)\), \(P(Q)\) and \(P(R)\), where \(Z\), \(Q\) and \(R\) are the additive groups of all integers, all rationals and all reals, respectively, with the natural linear order. In fact, \(P(Q)\) and \(P(R)\) are atoms in \([C_ 0, C^ 1]\); on the other hand, \(P(Z)\) fails to be an atom in \({\mathcal C}\) because \(P(Q)< P(Z)\).
Reviewer: F.Šik (Brno)

MSC:
06F15 Ordered groups
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