Definition/Summary A partially ordered set, or in short, a poset, is a set A together with a relation [tex]\leq~\subseteq A\times A[/tex] which is reflexive, antisymmetric, and transitive. In other words, satisfying 1)[itex]\forall x\in A,~x\leq x[/itex] (the relation is reflexive) 2)[itex]\forall x,y\in A,~x\leq y~and~y\leq x\Rightarrow x=y[/itex] (the relation is antisymmetric) 3)[itex]\forall x,y,z\in A,~x\leq y~and~y\leq z\Rightarrow x\leq z[/itex] (the relation is transitive) It is common to refer to a poset [itex]\left(A,\leq\right)[/itex] simply as A, with the underlying relation being implicit. Equations Antisymmetry: For example, a checker-board with the relation "not further from the white end" is not a poset because two different squares can be the same distance from the white end, and so the relation is not anti-symmetric. But the same board with the relation "not further from the white end according to legal moves for an ordinary black piece" is a poset . Totally ordered set: A poset with the extra condition 4) [itex]\forall x,y\in A,~x\leq y~or~y\leq x[/itex] is a totally ordered set. In other words, loosely speaking, a totally ordered set is essentially one-dimensional: it can be thought of as a line, in which any element is either one side or the other side of any other element. Extended explanation * This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!