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In
order theory
Order theory is a branch of mathematics that investigates the 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 intr ...
, a
subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
''A'' of a
partially ordered set
In mathematics, especially order theory, a partially ordered set (also poset) 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 binar ...
''P'' is a strong downwards antichain if it is an
antichain
In mathematics, in the area of order theory, an antichain is a subset of a partially ordered set such that any two distinct elements in the subset are incomparable.
The size of the largest antichain in a partially ordered set is known as its w ...
in which no two distinct elements have a common lower bound in ''P'', that is,
:
In the case where ''P'' is ordered by inclusion, and closed under subsets, but does not contain the empty set, this is simply a family of pairwise disjoint sets.
A strong upwards antichain ''B'' is a subset of ''P'' in which no two distinct elements have a common upper bound in ''P''. Authors will often omit the "upwards" and "downwards" term and merely refer to strong antichains. Unfortunately, there is no common convention as to which version is called a strong antichain. In the context of
forcing
Forcing may refer to: Mathematics and science
* Forcing (mathematics), a technique for obtaining independence proofs for set theory
*Forcing (recursion theory), a modification of Paul Cohen's original set theoretic technique of forcing to deal with ...
, authors will sometimes also omit the "strong" term and merely refer to antichains. To resolve ambiguities in this case, the weaker type of antichain is called a weak antichain.
If (''P'', ≤) is a partial order and there exist distinct ''x'', y ∈ ''P'' such that is a strong antichain, then (''P'', ≤) cannot be a
lattice (or even a Semilattice">meet semilattice
In mathematics, a join-semilattice (or upper semilattice) is a partially ordered set that has a join (mathematics), join (a least upper bound) for any nonempty set, nonempty finite set, finite subset. Duality (order theory), Dually, a meet-semilat ...
), since by definition, every two elements in a lattice (or meet semilattice) must have a common lower bound. Thus lattices have only trivial strong antichains (i.e., strong antichains of cardinality at most 1).
References
*{{Citation , last1=Kunen , first1=Kenneth , authorlink=Kenneth Kunen , title=Set Theory: An Introduction to Independence Proofs , url=https://archive.org/details/settheoryintrodu0000kune/page/53 , publisher=North-Holland Publishing Company , location=North Holland , series=Studies in logic and the foundations of mathematics , isbn=9780444854018 , year=1980 , pag
53
Order theory