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Lattices whose congruence lattices satisfy Lee’s identities. (English) Zbl 0771.06003

Generalizing trivial \(p\)-algebras, Boolean algebras and Stone algebras, \((L_ n)\)- and relative \((L_ n)\)-lattices are defined as equational subclasses of the class of all distributive \(p\)-algebras (or by suitable identities). \(\text{Con}(L)\) denotes the lattice of all the congruence relations of a lattice \(L\). Continuing papers of Grätzer, Haviar and Katríňak, the author characterizes lattices with \((L_ n)\)- and relative \((L_ n)\)-congruence lattices. In particular, different descriptions of lattices with Stone and relative Stone congruence lattices are given. These are simplified for weakly modular and semi- discrete lattices. Finally the distributive case is investigated. We give here some results:
Theorem 1. \(\text{Con}(L)\) is an \((L_ n)\)-lattice iff \(L\) is \(n\)-weakly modular and every \(n\)-tuple of mutually distinct congruence relations on \(L\) are \(n\)-weakly separable.
Theorem 3. Let \(L\) be a chain. Then \(\text{Con}(L)\) is an \((L_ n)\)- lattice iff \(L\) is discrete (i.e. \(\text{Con}(L)\) is a Boolean algebra).

MSC:

06B10 Lattice ideals, congruence relations
06B05 Structure theory of lattices
06D15 Pseudocomplemented lattices
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