ogposets.GrSubset

class rewalt.ogposets.GrSubset(support, ambient, **params)

Bases: object

Class for graded subsets, that is, pairs of a GrSet and an “ambient” OgPoset, where the first is seen as a subset of the second.

While objects of the class GrSet are mutable, once they are tied to an OgPoset they should be treated as immutable.

Parameters

support (GrSet) – The underlying graded set.
ambient (OgPoset) – The ambient oriented graded poset.

Keyword Arguments

wfcheck (bool) – Check whether the support is a well-formed subset of the ambient, that is, it has no elements out of range (default is True).

Notes

Two graded subsets are equal if and only if they have the same elements, and they are subsets of the same OgPoset.

Examples

We create an oriented graded poset and a pair of graded sets.

>>> point = OgPoset.point()
>>> triangle = point >> point >> point
>>> set1 = GrSet(El(1, 1), El(0, 1))
>>> set2 = GrSet(El(0, 3))

We can attach set1 to triangle as a subset.

>>> subset = GrSubset(set1, triangle)
>>> assert subset.support == set1

Trying to do the same with set2 returns a ValueError because El(0, 3) is out of range.

We can compute the downwards closure of set1 in triangle.

>>> subset.closure().support
GrSet(El(0, 0), El(0, 1), El(0, 2), El(1, 1))

All the set-theoretic operations apply to graded subsets as long as they have the same ambient OgPoset.

Methods

`closure`()	Returns the downwards closure of the graded subset.
`difference`(other)	Returns the difference with another graded subset of the same oriented graded poset.
`image`(ogmap)	Returns the image of the graded subset through a map of oriented graded posets.
`intersection`(*others)	Returns the intersection with other graded subsets of the same oriented graded poset.
`isdisjoint`(other)	Returns whether the object is disjoint from another graded subset of the same oriented graded poset.
`issubset`(other)	Returns whether the object is a subset of another subset of the same oriented graded poset.
`union`(*others)	Returns the union with other graded subsets of the same oriented graded poset.

Attributes

`ambient`	Returns the ambient oriented graded poset.
`dim`	Shorthand for `support.dim`.
`isclosed`	Returns whether the subset is (downwards) closed.
`support`	Returns the underlying graded set (the "support" of the subset).

property support

Returns the underlying graded set (the “support” of the subset).

Returns: support – The underlying graded set.
Return type: GrSet

property ambient

Returns the ambient oriented graded poset.

Returns: ambient – The ambient oriented graded poset.
Return type: OgPoset

property dim: Shorthand for support.dim.

property isclosed

Returns whether the subset is (downwards) closed.

Returns: isclosed – True if and only if the subset is downwards closed.
Return type: bool

union(*others)

Returns the union with other graded subsets of the same oriented graded poset.

Parameters: *others (GrSubset) – Any number of graded subsets of the same oriented graded poset.
Returns: union – The union of the graded subset with all the given others.
Return type: GrSubset

Notes

If all the arguments have type Closed, the union also has type Closed.

intersection(*others)

Returns the intersection with other graded subsets of the same oriented graded poset.

Parameters: *others (GrSubset) – Any number of graded subsets of the same oriented graded poset.
Returns: intersection – The intersection of the graded subset with all the given others.
Return type: GrSubset

Notes

If all the arguments have type Closed, the intersection also has type Closed.

difference(other)

Returns the difference with another graded subset of the same oriented graded poset.

Parameters: other (GrSubset) – Another graded subset of the same oriented graded poset.
Returns: difference – The difference between the two graded subsets.
Return type: GrSubset

issubset(other)

Returns whether the object is a subset of another subset of the same oriented graded poset.

Parameters: other (GrSubset) – Another graded subset of the same oriented graded poset.
Returns: issubset – True if and only self is a subset of other.
Return type: bool

isdisjoint(other)

Returns whether the object is disjoint from another graded subset of the same oriented graded poset.

Parameters: other (GrSubset) – Another graded subset of the same oriented graded poset.
Returns: issubset – True if and only self and other are disjoint.
Return type: bool

closure()

Returns the downwards closure of the graded subset.

Returns: closure – The downwards closure of the subset.
Return type: Closed

image(ogmap)

Returns the image of the graded subset through a map of oriented graded posets.

Parameters: ogmap (OgMap) – A map from the ambient to another OgPoset.
Returns: image – The image of the subset through the given map.
Return type: GrSubset

Notes

If the object has type Closed, its image has also type Closed.