In a
poset, a
chain is a
subset which is
totally ordered: any two
elements x and y are
comparable in the sense that either x ≤ y or y ≤ x (or both, in which case x = y, since a
partial order is
antisymmetric).
The idea is necessary because, in a partial order, not every element may be either greater or less than every other element. One prototypical example of a poset is the collection of subsets of a set X, where the partial order is the relation of set inclusion. Here it is possible for neither one of two subsets of X to contain the other.
For instance, in the poset of subsets of {1,2,3,4}, {{}, {1}, {1,4}, {1,3,4}, {1,2,3,4}} is a chain (in fact a maximal one, since no more subsets can be added which are comparable to all of these). {1,2} and {3,4} are an example of two elements which are not comparable to each other.
Paul Taylor, in his book Practical foundations of mathematics, notes that incomparability--failing to stand in the order relation either way around--"is what people usually mean by equality in politics".