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Keywords:
relatively normal lattice; algebraic lattice; complete distributivity; closed element; radical
relatively normal lattice; algebraic lattice; complete distributivity; closed element; radical
Summary:
Lattices in the class $\mathcal{IRN}$ of algebraic, distributive lattices whose compact elements form relatively normal lattices are investigated. We deal mainly with the lattices in $\mathcal{IRN}$ the greatest element of which is compact. The distributive radicals of algebraic lattices are introduced and for the lattices in $\mathcal{IRN}$ with the sublattice of compact elements satisfying the conditional join-infinite distributive law they are compared with two other kinds of radicals. Connections between complete distributivity of algebraic lattices and the distributive radicals are described. The general results can be applied e.g. to $MV$-algebras, $GMV$-algebras and unital $\ell $-groups.
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