- 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...
-
subset A of a
partially ordered set P is a
strong downwards antichain if it is an
antichain in
which no two
distinct elements have a
common lower bound...
-
satisfy the
countable chain condition, or to be ccc, if
every strong antichain in X is countable.
There are
really two conditions: the
upwards and downwards...
- on the size of an
antichain, the
sizes of the
largest antichain and of the
smallest chain decomposition are
again equal. An
antichain in a
partially ordered...
-
functions of n {\displaystyle n} variables. Equivalently, it is the
number of
antichains of
subsets of an n {\displaystyle n} -element set, the
number of elements...
- the two
additional values ± ∞ {\displaystyle \pm \infty } . If S is an
antichain (a set of
elements no two of
which are comparable) then the Dedekind–MacNeille...
- is that
every tree of
height ω1
either has a
branch of
length ω1 or an
antichain of
cardinality ℵ1. The
generalized Suslin hypothesis says that for every...
- inclusion.
Antichain principle:
Every partially ordered set has a
maximal antichain. Equivalently, in any
partially ordered set,
every antichain can be extended...
- sets is a
strict subset of
another is
called a
Sperner family, or an
antichain of sets, or a clutter. For example, the
family of k-element
subsets of...
- not an
antichain. In
other words, Fin ( E , 2 ) {\displaystyle \operatorname {Fin} (E,2)} -
antichains are countable. The
importance of
antichains in forcing...
