Generating sets for catalytic and projective distributive lattices
A distributive lattice \(L\) is said to be catalytic if \(\text{Hom} (L,L_ 1)\) is a lattice under pointwise order for all distributive lattices \(L_ 1\). The author has already given a characterization of catalytic lattices [Algebra Univers. 11, 334-340 (1980; Zbl 0457.06008)]: Let \(L\) be a distributive lattice. Then \(L\) is catalytic if and only if \(L\) and \(L^ d\), the dual lattice of \(L\), satisfy the following two conditions: (1) Every element of them is a join of finitely many join-irreducible elements; (2) The meet of two join-irreducible elements is again join- irreducible. The author provides another characterization of the catalytic lattices: Let \(L\) be a distributive lattice satisfying the conditions (1) and (2). Then \(L\) is catalytic if and only if the meet-semilattice \(J(L)\) of join- irreducible elements is generated and dually generated by the set \(I(J(L))\) of all irreducible elements of \(J(L)\). It is well known that the projective distributive lattices are catalytic. The converse is true for the countable case. In addition, the author shows that the uncountable projective distributive lattices satisfy some antichain conditions.
scientific article; zbMATH DE number 203153
scientific article