In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, especially in
order theory
Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article intr ...
, a
partially ordered set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set (mathematics), set. A poset consists of a set toget ...
with a unique
minimal element
In mathematics, especially in order theory, a maximal element of a subset ''S'' of some preordered set is an element of ''S'' that is not smaller than any other element in ''S''. A minimal element of a subset ''S'' of some preordered set is define ...
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
In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topolog ...
of a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
could be defined in terms of its
distributive lattice
In mathematics, a distributive lattice is a lattice in which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set uni ...
of
closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a cl ...
s. He observed that the inclusion order on the closed sets of a
T1 space
In topology and related branches of mathematics, a T1 space is a topological space in which, for every pair of distinct points, each has a neighborhood not containing the other point. An R0 space is one in which this holds for every pair of top ...
has the disjunction property. The generalization to partial orders was introduced by .
References
*.
*.
Order theory
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