Published online by Cambridge University Press: 20 November 2018
Throughout this paper when we refer to a semilattice S we shall mean that S is a meet semilattice. We shall denote the infimum of two elements a, b of S by a ∧ b, and the supremum, if it exists, by a ∨ b. A prime semilattice is a meet semilattice such that the infimum distributes over all existing finite suprema, in the sense that if x 1 ∨ x 2 … ∨ xn exists then (x ∧ x 1) ∨ (x ∧ x 2) … ∨ (x ∧ xn ) exists for any x and equals x ∧ (x 1 ∨ x 2 … ∨ xn ). Such semilattices were first studied by Balbes  and we use his terminology.
A non-empty subset F of S is a filter provided that x ∧ y ∊ F if and only if x ∊ F and y ∊ F.