# zbMATH — the first resource for mathematics

The lattice of R-subalgebras of a bounded distributive lattice. (English) Zbl 0542.06004
An R-subalgebra of a bounded, distributive lattice L is a sublattice that includes 0 and 1, and is closed under existing relative complements. The collection $$S_ R(L)$$ of all R-subalgebras of L is a lattice under inclusion. It is shown that $$S_ R(L)$$ is dually isomorphic to the lattice of all congruence relations (that is, kernels of morphisms) of the Priestley dual of L. This duality is then used to characterize those L such that $$S_ R(L)$$ is semimodular, modular, distributive, and Boolean. For example, it is shown that $$S_ R(L)$$ is distributive iff $$S_ R(L)$$ is Boolean iff L is a chain or a four element Boolean algebra.
Reviewer: R.S.Pierce

##### MSC:
 06D05 Structure and representation theory of distributive lattices 06D15 Pseudocomplemented lattices
Full Text: