On Banaschewski functions in lattices

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…