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RESEARCH PRODUCT

On Banaschewski functions in lattices

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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 to be antitonic globally. In some complemented lattices the complements can be chosen partially antitonic, that is, for any elements x and y with x y ' . This is possible for instance in semi-orthocomplemented lattices, see [6, (2.8), p. 7], and thus also in complemented modular lattices, see [6, (3.9) and (3.10), p. 12] or [ 10, Theorem 5]. In order to guarantee the existence of a B-function on a subset X of L we consider after the preliminaries three groups of assumptions concerning the elements of the lattice L" complementary properties, join-continuity, and covering