Strong antichain

In order theory, a subset A of a partially ordered set X is a strong downwards antichain if it is an antichain in which no two distinct elements have a common lower bound, that is,

$\forall x,y\in A\;[x\neq y\rightarrow \neg \exists z\in A\;[z\leq x\land z\leq y]].$

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.

References

• Kunen, Kenneth (1980), Set Theory: An Introduction to Independence Proofs (PDF), Studies in logic and the foundations of mathematics, North Holland: North-Holland Publishing Company, p. 53, ISBN 9780444854018