In the
mathematical
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
field of
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 ...
, an inclusion order is the
partial order that arises as the
subset
In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), 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 a ...
-inclusion relation on some collection of objects. In a simple way, every
poset ''P'' = (''X'',≤) is (
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
to) an inclusion order (just as every group is isomorphic to a
permutation group – see
Cayley's theorem
In the mathematical discipline of group theory, Cayley's theorem, named in honour of Arthur Cayley, states that every group is isomorphic to a subgroup of a symmetric group.
More specifically, is isomorphic to a subgroup of the symmetric gro ...
). To see this, associate to each element ''x'' of ''X'' the set
:
then the transitivity of ≤ ensures that for all ''a'' and ''b'' in ''X'', we have
:
There can be sets
of
cardinality
The thumb is the first digit of the hand, next to the index finger. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thum ...
less than
such that ''P'' is
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
to the inclusion order on ''S''. The size of the smallest possible ''S'' is called the
2-dimension of ''P''.
Several important classes of poset arise as inclusion orders for some natural collections, like the
Boolean lattice ''Q''
''n'', which is the collection of all 2
''n'' subsets of an ''n''-element set, the interval-containment orders, which are precisely the orders of
order dimension at most two, and the dimension-''n'' orders, which are the containment orders on collections of ''n''-boxes anchored at the
origin. Other containment orders that are interesting in their own right include the circle orders, which arise from disks in the plane, and the angle orders.
See also
*
Birkhoff's representation theorem
*
Intersection graph
*
Interval order
References
*
*
Order theory
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