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".