1.Introduction. An implicative semi-lattice is an algebraic system having as models logical systems equipped with implication and conjunction, but not possessing a disjunction. The position of implicative semi-lattices in algebraic logic is clearly displayed in [2]. In [1], the relation of implicative lattices to Brouwerian logics is explained. In the terminology of [1], an implicative lattice would be called a relatively pseudo-complemented lattice. Monteiro [6] and Glivenko [5] have each given a set of axioms for Brouwerian logic. Glivenko has also proved certain results for Brouwerian logics (equipped with disjunction) which are proved for implicative semi-lattices in §4 of this paper. In [3] and [7], the relation of implicative lattices to closure algebras and topologies is explored.There are four principal results in this paper. The first (Theorem 3.2) states that the relationship between homomorphisms of implicative semi-lattices and their kernels is that usually found in algebra, and not found in lattice theory, namely, that every homomorphism is essentially determined by its kernel. In [4], Frink has shown that the structure of a pseudo-complemented semi-lattice, and therefore of a bounded implicative semi-lattice, is very similar to that of a bounded implicative lattice, in that the semi-lattice of "closed" elements is a Boolean algebra, despite the absence of a least upper bound. The second result of this paper (Theorem 4.4) states that every element of a bounded implicative semi-lattice is the meet of its closure with a dense element, a result which is well known for bounded implicative lattices. The third result (Theorem 5.2) gives a method for constructing all bounded implicative semi-lattices having a given Boolean algebra for closed algebra, and a given implicative semi-lattice for dense filter; the fourth result (Theorem 6.6) gives essentially the same information about homomorphisms.

WILLIAM C. NEMITZ(') 1. Introduction. An implicative semi-lattice is an algebraic system having as models logical systems equipped with implication and conjunction, but not possessing a disjunction. The position of implicative semi-lattices in algebraic logic is clearly displayed in [2]. In [1], the relation of implicative lattices to Brouwerian logics is explained. In the terminology of [1], an implicative lattice would be called a relatively pseudo-complemented lattice. Monteiro [6] and Glivenko [5] have each given a set of axioms for Brouwerian logic. Glivenko has also proved certain results for Brouwerian logics (equipped with disjunction) which are proved for implicative semi-lattices in §4 of this paper. In [3] and [7], the relation of implicative lattices to closure algebras and topologies is explored. There are four principal results in this paper. The first (Theorem 3.2) states that the relationship between homomorphisms of implicative semi-lattices and their kernels is that usually found in algebra, and not found in lattice theory, namely, that every homomorphism is essentially determined by its kernel. In [4], Frink has shown that the structure of a pseudo-complemented semi-lattice, and therefore of a bounded implicative semi-lattice, is very similar to that of a bounded implicative lattice, in that the semi-lattice of "closed" elements is a Boolean algebra, despite the absence of a least upper bound. The second result of this paper (Theorem 4.4) states that every element of a bounded implicative semi-lattice is the meet of its closure with a dense element, a result which is well known for bounded implicative lattices. The third result (Theorem 5.2) gives a method for constructing all bounded implicative semi-lattices having a given Boolean algebra for closed algebra, and a given implicative semi-lattice for dense filter; the fourth result (Theorem 6.6) gives essentially the same information about homomorphisms.

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