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Closure operators on radical classes of lattice-ordered groups. (English) Zbl 0661.06007
The author studies extensions of one $$\ell$$-group A by another $$\ell$$- group B. He 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. This closure property is discussed. Theorems 3.5 and 3.8 say that the closed-kernel radical classes form a complete lattice under inclusion and are subsemigroups 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 typical 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 $$(R\cdot S)^ h\subset R^ h\cdot S^ h$$, 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
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