Relative annihilators in semilattices

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Abstract
An α-distributive (respectively α-implicative) semilattice S is a lower semilattice (with greatest lower bound denoted by juxtaposition) in which the annihilator 〈x, a〉, that is {y ∈ S: xy ≤ α}, is an ideal (respectively a principal ideal) for the fixed element α and any x of S. These semilattices appear as natural links between general and distributive semi-lattices on the one hand, and between pseudo-complemented and implicative semilattices on the other hand. Prime and dense elements, as well as maximal and prime filters, are essential. Mandelker's result, a lattice L is distributive if and only if 〈x, y〉 is an ideal for any x, y ∈ L is extended to semi-lattices.

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