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**Closure operators on radical classes of lattice-ordered groups.**
*(English)*
Zbl 0624.06022

Let A and B be \(\ell\)-groups. The author defines the upper product of A by B on the cartesian product \(A\times B\), special cases of which are e.g. cardinal sums, wreath products and lex extensions and adds one more, the so called P-product and gives some properties of it. For some radical classes (e.g. for any K-radical class) R the radical kernel R(G) is a closed \(\ell\)-ideal of G, thus there exists a minimal closed-kernel radical class \(R^ c\) containing R. There is discussed this closure property. Theorems 3.5 and 3.8 say that the closed-kernel radical classes form a complete lattice under inclusion and are a subsemigroup of the radical classes. Analogously as before there are defined new radical classes from old ones by taking the double polar of the kernels. This yields the class of radical classes whose kernels are always polars and the p-closure operator.

Some representative results. Theorem 4.3. The lattice of radical classes is a pseudocomplemented lattice whose skeletal elements are precisely those radical classes with polar kernels. Theorem 5.1. For any radical class R there exist unique minimal radical classes \(R^ s\) and \(R^ h\), closed with respect to \(\ell\)-subgroups and \(\ell\)-homomorphic images, respectively, that contain R. Moreover, the collections of s-closed and h-closed radical classes form complete lattices under inclusion. Theorem 5.7. For any two radical classes R and S, \((R^ h\cdot S^ h)^ h=R^ h\cdot S^ h\) and \((R\cdot S)^ h\subseteq R^ h\cdot S^ h\), where \(R\cdot S\) denotes the product of radical classes R and S defined as the class of \(\ell\)-groups G such that \(S(G/R(G))=G/R(G)\). The author shows that usually the four closure operators c,p,s,h do not commute with one another.

Some representative results. Theorem 4.3. The lattice of radical classes is a pseudocomplemented lattice whose skeletal elements are precisely those radical classes with polar kernels. Theorem 5.1. For any radical class R there exist unique minimal radical classes \(R^ s\) and \(R^ h\), closed with respect to \(\ell\)-subgroups and \(\ell\)-homomorphic images, respectively, that contain R. Moreover, the collections of s-closed and h-closed radical classes form complete lattices under inclusion. Theorem 5.7. For any two radical classes R and S, \((R^ h\cdot S^ h)^ h=R^ h\cdot S^ h\) and \((R\cdot S)^ h\subseteq R^ h\cdot S^ h\), where \(R\cdot S\) denotes the product of radical classes R and S defined as the class of \(\ell\)-groups G such that \(S(G/R(G))=G/R(G)\). The author shows that usually the four closure operators c,p,s,h do not commute with one another.

Reviewer: F.Šik

### MSC:

06F15 | Ordered groups |

06B23 | Complete lattices, completions |

20F60 | Ordered groups (group-theoretic aspects) |

### Keywords:

\(\ell \)-groups; upper product; P-product; radical classes; radical kernel; closed-kernel radical class; polars; p-closure operator; lattice of radical classes; pseudocomplemented lattice; skeletal elements; product of radical classes
Full Text:
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### References:

[1] | Ball R., Conrad P., and Darnel M.: Above and Below Subgroups of a Lattice-ordered Group. submitted to Transactions of the American Mathematical Society. A summary of the principal results appears in Ordered Algebraic Structures, published 1985 by Marcel Dekker, edited Wayne Powell and Constantine Tsinakis. · Zbl 0577.06015 |

[2] | Bigard A., Keimel K., and Wolfenstein S.: Groupes et Anneaux Réticulés. Lecture Notes in Mathematics 608, Springer Verlag, 1977. · Zbl 0384.06022 |

[3] | Birkhoff G.: Lattice Theory. AMS Colloquium Publications 25, 1979. · Zbl 0505.06001 |

[4] | Bixler J. P., Darnel M.: Special-valued l-groups. to appear in Algebra Universalis. · Zbl 0582.06017 |

[5] | Bleier R., Conrad P.: The Lattice of Closed l-idelas and a*-extensions of an Abelian l-group. Pacific Journal of Mathematics, 47 (2) (1973), 329-340. · Zbl 0238.06012 |

[6] | Conrad P.: Lattice-ordered Groups. Tulane Lecture Notes, Tulane University, 1970. · Zbl 0258.06011 |

[7] | Conrad P.: The Essential Closure of an Archimedean l-group. Duke Math. J., 38 (1971), 151-160. · Zbl 0216.03104 |

[8] | Conrad P.: Epi-archimedean l-groups. Czech. Math. J., 24 (99) (1974), 192-218. · Zbl 0319.06009 |

[9] | Conrad P.: Torsion Radicals of Lattice-ordered Groups. Symposia Mathematica 21, Academic Press, 1977, 479-513. · Zbl 0372.06011 |

[10] | Conrad P.: K-radical Classes of Lattice-ordered Groups. Algebra Carbondale 1980, Lecture Notes in Mathematics 848, Springer-Verlag, 1980. |

[11] | Holland W. C: The Lattice-ordered Group of Automorphisms of an Ordered Set. Michigan Math. J., 10 (1963), 399-408. · Zbl 0116.02102 |

[12] | Hollnad W. C: Varieties of l-groups are Torsion Classes. Czech. Math. J., 29 (104) (1979), 11-12. · Zbl 0432.06011 |

[13] | Holland W. C., McCleary S.: Wreath Products of Ordered Permutation Groups. Pacific J. Math., 31 (1969), 703-716. · Zbl 0206.31804 |

[14] | Jakubík J.: Radical Mappings and Radical Classes of Lattice-ordered Groups. Symposia Mathematica 21, Academic Press, 1977, 451-477. |

[15] | Jakubík J.: Products of Radical Classes of Lattice-ordered Groups. Acta Mathematica Comnenianae, 39 (1980), 31 - 41. |

[16] | Kenny G. O.: Lattice-ordered Groups. PhD dissertation. University of Kansas, 1975. |

[17] | Martinez J.: Varieties of Lattice-ordered Groups. Math. Zeit., 137 (1974), 265-284. · Zbl 0274.20034 |

[18] | Martinez J.: Torsion Theory for l-groups, I. Czech. Math. J., 25 (100) (1975), 284-294. |

[19] | Read J.: Wreath Products of Nonoverlapping Lattice-ordered Groups. Can. Math. Bull., 17 (5) (1975), 713-722. · Zbl 0313.06012 |

[20] | Walker R.: The Stone-Cech Compactification, Ergebnisse der Mathematik und ihrer Grenzgebiete. Band 83, Springer-Verlag, 1977. |

[21] | Weinberg E. C: Free Lattice-ordered Abelian Groups. Math. Ann., 151 (1963), 187-199. · Zbl 0114.25801 |

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