The characterization of -compact elements in some lattices
We attach to each -semilattice a graph whose vertices are join-irreducible elements of and whose edges correspond to the reflexive dependency relation. We study properties of the graph both when is a join-semilattice and when it is a lattice. We call a -semilattice particle provided that the set of its join-irreducible elements satisfies DCC and join-generates . We prove that the congruence lattice of a particle lattice is anti-isomorphic to the lattice of all hereditary subsets of...