Strong antichain

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,

:$forall\; x,\; y\; in\; A\; mbox\{such\; that\; \}\; x\; ot=\; y\; otexists\; z\; x\; geq\; z\; and\; y\; geq\; z.$

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.