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.
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.
[1] The generalization to partial orders was introduced by Wolk (1956).
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