An algebraic approach to partial lattices. (English) Zbl 0910.06006

Chajda, I.; Seidl, Z.

Demonstr. Math. 30, No. 3, 485-494 (1997).

In partial lattices the concepts of distributivity and modularity split up in various ways. The authors consider some of these concepts including what they call strongly modular, weakly modular, extensibly modular, weakly distributive, strongly distributive, and extensibly distributive. They characterize all these kinds of distributivity and modularity by forbidden partial sublattices. For example, they show that a partial lattice is weakly modular if and only if it does not contain a partial sublattice looking like the pentagon (Theorem 5) and that a partial sublattice is weakly distributive if and only if it does not contain a partial sublattice looking “like” the pentagon or the diamond (Theorem 8).

Reviewer: M.Stern (Halle)

MSC:

06B99	Lattices
06A06	Partial orders, general

Keywords:

partial lattices; distributivity; modularity