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partially ordered

 Definition/Summary A partially ordered set, or in short, a poset, is a set A together with a relation $$\leq~\subseteq A\times A$$ which is reflexive, antisymmetric, and transitive. In other words, satisfying 1)$\forall x\in A,~x\leq x$ (the relation is reflexive) 2)$\forall x,y\in A,~x\leq y~and~y\leq x\Rightarrow x=y$ (the relation is antisymmetric) 3)$\forall x,y,z\in A,~x\leq y~and~y\leq z\Rightarrow x\leq z$ (the relation is transitive) It is common to refer to a poset $\left(A,\leq\right)$ simply as A, with the underlying relation being implicit.

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 Breakdown Mathematics > Foundations >> Set Theory

 Extended explanation 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) $\forall x,y\in A,~x\leq y~or~y\leq x$ 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.