Semi-Boolean lattices.

Abstract

An implicative semi-lattice is an algebraic system (L, <, Λ, *) in which (L, <, Λ) is a meet semi-lattice, and * is a binary composition such that x ^ y * z if and only if x Λ y< z for all elements x, y, z, of L. Every implicative semi-lattice has a greatest element, denoted by 1.If an implicative semi-lattice has a least element 0, then it is called bounded.In a bounded implicative semi-lattice L, elements of the form x * 0 are called "closed".The set of closed elements forms a Boolean algebra which is a sub-implicative semi-lattice of L but not necessarily a sub-lattice.By a sub-lattice of an implicative semi-lattice we shall mean a sub-implicative semi-lattice which is a lattice and such that the join of any two elements of the sub-lattice is also a join in the semi-lattice.An implicative lattice is simply an implicative semi-lattice which happens to be a lattice.Birkhoff [l] identifies bounded implicative lattices with Brouwerian logics.In general, the join of an implicative lattice is not very closely related to the implication.An exception to this is the case of a Boolean algebra.Here the join of two elements a and b always equals the element (a * b) * b.With this as a starting point, we make the following definition.Definition 1.By the pseudo-join ab, of two elements a and b of an implicative semi-lattice Z,, we shall mean the element {(a * b) * b) Λ ((b * a) * a).Theorem 1.Let L be an implicative semi-lattice, and let a and b be elements of L. Then (1) α< ab, δ < ab (2) a < b if and only if ab = b (3) aa = a 9 ab = ba