On Banaschewski functions in lattices

1991

hold for all x, y ~ X. We call such a function z a Banaschewski function or a B-function on X. A lattice L is a B-lattice or antitonely complemented, if there is a B-function defined on the whole lattice L. For instance, Boolean lattices as well as orthocomplemented lattices are B-lattices. On the other hand, a B-lattice is not necessarily Boolean or orthocomplemented, although a distributive B-lattice is a Boolean lattice. It is shown later that a matroid (geometric) lattice is also a B-lattice. Naturally, our results include the lemma of Banaschewski [ 1, Lemma 4], by which the lattice of the subspaces of a vector space is a B-lattice. It should be emphasized that a B-function is supposed…

Counterexamples to the Algebraic Closed Graph Theorem

1982

Disjointness of Lattice Elements

1992

We examine the relations between various disjointness properties in lattices with least elements, and in special lattices like section semicomplemented lattices and section complemented lattices.