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 Wallman (1938), 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 Wolk (1956).

References

• Wallman, Henry (1938), "Lattices and topological spaces", Annals of Mathematics, 39 (1): 112–126, doi:10.2307/1968717, JSTOR 0003486.
• Wolk, E. S. (1956), "Some Representation Theorems for Partially Ordered Sets", Proceedings of the American Mathematical Society, 7 (4): 589–594, doi:10.2307/2033355, JSTOR 00029939.