On irreducible elements of semilattices

The concept of (join)-irreducible elements works well, especially for distributive lattices. Therefore our definition of elements of a given degree of irreducibility employs the notion of distributivity as much as possible, even if the irreducibility is defined for elements of a (meet)-semilattice. Via the lattice of hereditary subsets of the poset ofk-irreducible elements of a semilattice (wherek is a cardinal) we obtain a new construction of a D1_k-reflection (a sort of free distributive extension) of the semilattice, provided that there are sufficiently manyk-irreducible elements. The last property is satisfied, for example, if the original semilattice is the dual of an algebraic lattice Dilworth and Crawley, 1960], but this condition is too restrictive for semilattices. It turns out that, under certain limitations, the D1_k-reflection of a semilattice both preserves and reflects the degree of irreducibility.Presented by R. Freese.