## 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.
Reviewer: F.Šik

### MSC:

 06F15 Ordered groups 06B23 Complete lattices, completions 20F60 Ordered groups (group-theoretic aspects)
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