Revision history [back]

First, note that your list U is not a list of vectors but a list of lists. So, the first thing to do is to transform it into a list of vectors (forget about the immutable=True for now):

sage: V = [vector(u, immutable=True) for u in U]
sage: V
[(1, 0, 0, 0, 0),
 (0, 0, 0, 1, 1),
 (1, 1, 0, 0, 0),
 (0, 0, 1, 0, 0),
 (0, 0, 0, 1, 0),
 (0, 1, 0, 0, 1),
 (1, 0, 1, 0, 0),
 (0, 0, 0, 0, 1),
 (1, 0, 1, 1, 0),
 (0, 1, 0, 0, 0),
 (0, 0, 1, 1, 1),
 (1, 1, 1, 1, 1)]

Now, if you look at the Poset constructor documentation:

sage: Poset?

you will see that the second way to build a poset is to provide a pair (E,f) where E are the elements of the poset and f(x,y) is a function that returns True when x<=y in the poset. This corresponds to your description.

So we can do:

sage: P = Poset((V, lambda v,w: w-v in V))

where lambda v,w: w-v in V s a Python way to describe "the map that, given v and w, returns whether w-v belongs to V".

You can check that you have the expected poset by drawing it:

sage: P.show()

Remark: everything seems to be very natural (we just translated your question into Sage/Python), except perhaps the immutable=True. This is due to the fact that, if the elements of V are mutable, you will get the following error:

sage: V = [vector(u) for u in U]
sage: P = Poset((V, lambda v,w: w-v in V))
TypeError: mutable vectors are unhashable

First, note that your list U is not a list of vectors but a list of lists. So, the first thing to do is to transform it into a list of vectors (forget about the immutable=True for now):

sage: V = [vector(u, immutable=True) for u in U]
sage: V
[(1, 0, 0, 0, 0),
 (0, 0, 0, 1, 1),
 (1, 1, 0, 0, 0),
 (0, 0, 1, 0, 0),
 (0, 0, 0, 1, 0),
 (0, 1, 0, 0, 1),
 (1, 0, 1, 0, 0),
 (0, 0, 0, 0, 1),
 (1, 0, 1, 1, 0),
 (0, 1, 0, 0, 0),
 (0, 0, 1, 1, 1),
 (1, 1, 1, 1, 1)]

Now, if you look at the Poset constructor documentation:

sage: Poset?

you will see that the second way to build a poset is to provide a pair (E,f) where E are the elements of the poset and f(x,y) is a function that returns True when x<=y in the poset. This corresponds to your description.

So we can do:

sage: P = Poset((V, lambda v,w: w-v in V))

where lambda v,w: w-v in V s is a Python way to describe "the map that, given v and w, returns whether w-v belongs to V".

You can check that you have the expected poset by drawing it:

sage: P.show()

Remark: everything seems to be very natural (we just translated your question into Sage/Python), except perhaps the immutable=True. This is due to the fact that, if the elements of V are mutable, you will get the following error:

sage: V = [vector(u) for u in U]
sage: P = Poset((V, lambda v,w: w-v in V))
TypeError: mutable vectors are unhashable