Strong antichain

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In order theory, a subset A of a partially ordered set X is said to be a strong downwards antichain if no two elements have a common lower bound, that is,

\forall x, y \in A \ \ \mbox{such that } x \not= y \ \not\exists z \ \ x \geq z \ and \ y \geq z.

A strong upwards antichain is defined similarly.

Often authors will drop the upwards/downwards term and merely refer to strong antichains. Unfortunately, there is no common convention as to which version is called a strong antichain.

Because it is convenient to have a convention, and because the two are essentially equivalent concepts, we will adopt the convention that a strong antichain means a strong downwards antichain.

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