Disjunction property of Wallman

In mathematics, especially in order theory, a partially ordered set with a unique minimal element 0 has the disjunction property of Wallman when for every pair ("a", "b") of elements of the poset, either "b" ≤ "a" or there exists an element "c" ≤ "b" such that "c" ≠ 0 and "c" has no nontrivial common predecessor with "a". That is, in the latter case, the only "x" with "x" ≤ "a" and "x" ≤ "c" is "x" = 0.
A version of this property for lattices was introduced by , in a paper showing that the homology theory of a topological space could be defined in terms of its distributive lattice of closed sets. He observed that the inclusion order on the closed sets of a T1 space has the disjunction property. The generalization to partial orders was introduced by .