- 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...
- 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...
-
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...
-
monotone Boolean functions of n variables. Equivalently, it is the
number of
antichains of
subsets of an n-element set, the
number of
elements in a free distributive...
- 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...
-
ordered set in
terms of a
partition of the
order into a
minimum number of
antichains. It is
named for Leon Mirsky (1971) and is
closely related to Dilworth's...
- inclusion.
Antichain principle:
Every partially ordered set has a
maximal antichain. Equivalently, in any
partially ordered set,
every antichain can be extended...
- to
indicate a
related stronger notion; for example, a
strong antichain is an
antichain satisfying certain additional conditions, and
likewise a strongly...
-
every element of P has a
lower bound in D. For
application of ccc, an
antichain is a
subset A of P such that any two
distinct members of A are incompatible...
