Direct limits of cyclically ordered groups.

*(English)*Zbl 0821.06015The 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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\textit{J. Jakubík} and \textit{G. Pringerová}, Czech. Math. J. 44, No. 2, 231--250 (1994; Zbl 0821.06015)

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##### References:

[1] | C.G. Chehata: An algebraically simple ordered group. Proc. London Math. Soc. 2 (1952), 183-197. · Zbl 0046.02501 |

[2] | C.G. Chehata, R. Wiegandt: Radical theory for fully ordered groups. Mathematica - Rév. d’Anal. Numér. Théor. Approx. 20 (1979), no. 43, 143-157. · Zbl 0409.06008 |

[3] | V Dlab: On a family of simple ordered groups. J. Austral. Math. Soc. 8 (1968), 591-608. · Zbl 0165.34502 |

[4] | B.J. Gardner: Some aspects of radical theory for fully ordered groups. Comment. Math. Univ. Carol. 26 (1985), 821-837. · Zbl 0584.06010 |

[5] | D. Gluschankof: Cyclic ordered groups and \(MV\)-algebras. Czechoslovak Math. J. 43(118) (1993), 249-263. · Zbl 0795.06015 |

[6] | G. Grätzer: Universal Algebra. Princeton, 1968. · Zbl 0182.34201 |

[7] | E. Halušková: Direct limits of monounary algebras (to be submitted). · Zbl 1004.08003 |

[8] | G. Higman, A.H. Stone: On inverse systems with trivial limits. J. London Math. Soc. 29 (1954), 233-236. · Zbl 0055.02503 |

[9] | P.D. Hill: Relation of a direct limit group to associated vector group. Pacif. J. Math. 10 (1960), 1309-1312. · Zbl 0102.26303 |

[10] | P.D. Hill: Note on a direct limit group. Amer. Math. Monthly 67 (1960), 998-1000. · Zbl 0111.24302 |

[11] | P.D. Hill: Limits of sequences of finitely generated abelain groups. Proc. Amer. Math. Soc. 12 (1961), 946-950. · Zbl 0102.26702 |

[12] | J. Jakubík: On the lattice of radical classes of linearly ordered groups. Studia scient. math. Hungar 19 (1981), 76-86. |

[13] | J. Jakubík: Retracts of abelian cyclically ordered groups. Archivum mathem. 25 (1989), 13-18. · Zbl 0712.06013 |

[14] | J. Jakubík, G. Pringerová: Representations of cyclically ordered groups. Časopis. pěst. matem. 113 (1988), 184-196. · Zbl 0654.06016 |

[15] | J. Jakubík, G. Pringerová: Radical classes of cyclically ordered groups. Math. Slovaca 38 (1988), 255-268. · Zbl 0662.06004 |

[16] | A.G. Kurosh: Group theory, third edition. Moskva, 1967. · Zbl 0189.30801 |

[17] | V.M. Kopytov: Lattice ordered groups. Moskva, 1984. · Zbl 0567.06011 |

[18] | L. Rieger: On ordered and cyclically ordered groups, I, II, III. Věstník Král. čes. spol. nauk (1946), 1-31. |

[19] | S. Swierczkowski: On cyclically ordered groups. Fund. Math. 47 (1959), 161-166. · Zbl 0096.01501 |

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