ogposets.Closed

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

Bases: GrSubset

Subclass of GrSubset for (downwards) closed subsets.

Implements a number of methods that do not make sense for non-closed subsets, in particular those computing input and output boundaries in each dimension.

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, closed subset of the ambient (default is True).

Notes

There is an alternative constructor subset() which takes a GrSubset, and “upgrades” it to a Closed if it is downwards closed.

Examples

After creating an oriented graded poset, we can obtain the closed subset of all its elements with OgPoset.all().

>>> point = OgPoset.point()
>>> triangle = point >> point >> point
>>> all = triangle.all()

We can compute its input and output boundary…

>>> all_in = all.input
>>> all_out = all.output

And since all happens to be a molecule, we can check the “globular” relations.

>>> assert all_in.input == all_out.input
>>> assert all_in.output == all_out.output

Methods

`boundary`([sign, dim])	Returns the boundary of a given orientation and dimension.
`boundary_max`([sign, dim])	Returns the subset of maximal elements of the boundary of a given orientation and dimension.
`maximal`()	Returns the subset of maximal elements, that is, those that are not covered by any other element in the closed subset.
`subset`(grsubset, **params)	Alternative constructor that promotes a `GrSubset` to a `Closed`.

Attributes

`as_map`	Returns an injective map representing the inclusion of the closed subset in the ambient.
`input`	Alias for `boundary('-')`.
`ispure`	Returns whether the maximal elements of the closed subset all have the same dimension.
`isround`	Returns whether the closed subset is round ("has spherical boundary").
`output`	Alias for `boundary('+')`.

property as_map

Returns an injective map representing the inclusion of the closed subset in the ambient.

Returns: as_map – A map of oriented graded posets representing the inclusion of the closed subset.
Return type: OgMap

property ispure

Returns whether the maximal elements of the closed subset all have the same dimension.

Returns: ispure – True if and only if the subset is pure.
Return type: bool

property isround

Returns whether the closed subset is round (“has spherical boundary”).

This means that, for all k smaller than the dimension of the subset, the intersection of its input k-boundary and of its output k-boundary is equal to its (k-1)- boundary.

Returns: isround – True if and only if the subset is round.
Return type: bool

maximal()

Returns the subset of maximal elements, that is, those that are not covered by any other element in the closed subset.

Returns: maximal – The subset of maximal elements.
Return type: GrSubset

boundary_max(sign=None, dim=None)

Returns the subset of maximal elements of the boundary of a given orientation and dimension.

Parameters

sign (str, optional) – Orientation: '-' for input, '+' for output, None (default) for both.
dim (int, optional) – Dimension of the boundary (default is self.dim - 1).

Returns

boundary_max – The maximal elements of the requested boundary.

Return type

GrSubset

boundary(sign=None, dim=None)

Returns the boundary of a given orientation and dimension.

Parameters

sign (str, optional) – Orientation: '-' for input, '+' for output, None (default) for both.
dim (int, optional) – Dimension of the boundary (default is self.dim - 1).

Returns

boundary – The requested boundary subset.

Return type

Closed

property input: Alias for boundary('-').

property output: Alias for boundary('+').

static subset(grsubset, **params)

Alternative constructor that promotes a GrSubset to a Closed.

Parameters: grsubset (GrSubset) – The subset to promote.
Keyword Arguments: wfcheck (bool) – Check whether the subset is downwards closed (default is True).