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Radicals and complete distributivity in relatively normal lattices. (English) Zbl 1052.06009
Summary: Lattices in the class IRN of algebraic distributive lattices whose compact elements form relatively normal lattices are investigated. We deal mainly with the lattices in IRN the greatest element of which is compact. The distributive radicals of algebraic lattices are introduced and for the lattices in 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.
MSC:
06D15 Pseudocomplemented lattices
06D20 Heyting algebras (lattice-theoretic aspects)
06D35 MV-algebras
06F05 Ordered semigroups and monoids
06F15 Ordered groups
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