In order theory

Order theory

Order theory is a branch of mathematics which investigates our intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and gives some basic definitions...

, a subset

Subset

In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...

 A of a partially ordered set

Partially ordered set

In mathematics, especially order theory, a partially ordered set formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation that indicates that, for certain pairs of elements in the...

 X is said to be a strong downwards antichain if no two elements have a common lower bound, that is,


A strong upwards antichain is defined similarly.

A strong antichain is an antichain

Antichain

In mathematics, in the area of order theory, an antichain is a subset of a partially ordered set such that any two elements in the subset are incomparable. Let S be a partially ordered set...

.

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.