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Concerning your first question, you can see how to construct a custom poset by typing:

sage: Poset?

and look for the examples. There are also some pre-built posets in Sage, you can get the list by typing:

sage: posets.<TAB>

where <TAB> stands for the tabulation.

Concerning your second question, if L denotes the list you constructed, you can build the poset of its elements (viewed as subsets of ${1, 2, 4, 3, 6, 12}$) ordered by the inclusion as follows:

sage: PP  = Poset(([Set(s) for s in L], attrcall("issubset")))

Concerning your third question, the parapeter of the .order_ideals() method is a list of elements of your poset. It returns the elements of P that are smaller of some element of the given list.

In your case:

sage: P.order_ideal([12])
[1, 2, 4, 3, 6, 12]
sage: P.order_ideal([4]) 
[1, 2, 4]
sage: P.order_ideal([6])
[1, 2, 3, 6]
sage: P.order_ideal([6,4])
[1, 2, 4, 3, 6]
sage: P.order_ideal([])
[]

Concerning your first question, you can see how to construct a custom poset by typing:

sage: Poset?

and look for the examples. at the EXAMPLES: section. There are also some pre-built posets in Sage, you can get the list by typing:

sage: posets.<TAB>

where <TAB> stands for the tabulation.

Concerning your second question, if L denotes the list you constructed, you can build the poset of its elements (viewed as subsets of ${1, 2, 4, 3, 6, 12}$) ordered by the inclusion as follows:

sage: PP  = Poset(([Set(s) for s in L], attrcall("issubset")))

Concerning your third question, the parapeter of the .order_ideals() method is a list of elements of your poset. It returns the elements of P that are smaller of some element of the given list.

In your case:

sage: P.order_ideal([12])
[1, 2, 4, 3, 6, 12]
sage: P.order_ideal([4]) 
[1, 2, 4]
sage: P.order_ideal([6])
[1, 2, 3, 6]
sage: P.order_ideal([6,4])
[1, 2, 4, 3, 6]
sage: P.order_ideal([])
[]

Concerning your first question, you can see how to construct a custom poset by typing:

sage: Poset?

and look at the EXAMPLES: section. There are also some pre-built posets in Sage, you can get the list by typing:

sage: posets.<TAB>

where <TAB> stands for the tabulation.

Concerning your second question, if L denotes the list you constructed, you can build the poset of its elements (viewed as subsets of ${1, 2, 4, 3, 6, 12}$) ordered by the inclusion as follows:

sage: PP  = Poset(([Set(s) for s in L], attrcall("issubset")))

Concerning your third question, the parapeter of the .order_ideals() method is a list of elements of your poset. It returns the elements of P that are smaller of some element of the given list.

In your case:

sage: P.order_ideal([12])
[1, 2, 4, 3, 6, 12]
sage: P.order_ideal([4]) 
[1, 2, 4]
sage: P.order_ideal([6])
[1, 2, 3, 6]
sage: P.order_ideal([6,4])
[1, 2, 4, 3, 6]
sage: P.order_ideal([])
[]

Concerning your first question, you can see how to construct a custom poset by typing:

sage: Poset?

and look at the EXAMPLES: section. There are also some pre-built posets in Sage, you can get the list by typing:

sage: posets.<TAB>

where <TAB> stands for the tabulation.

Concerning your second question, if L denotes the list you constructed, you can build the poset of its elements (viewed as subsets of ${1, {1, 2, 4, 3, 6, 12}$) 12}) ordered by the inclusion as follows:

sage: PP  = Poset(([Set(s) for s in L], attrcall("issubset")))

Concerning your third question, the parapeter parameter of the .order_ideals() method is a list of elements of your poset. It returns the elements of P that are smaller of some element of the given list.

In your case:

sage: P.order_ideal([12])
[1, 2, 4, 3, 6, 12]
sage: P.order_ideal([4]) 
[1, 2, 4]
sage: P.order_ideal([6])
[1, 2, 3, 6]
sage: P.order_ideal([6,4])
[1, 2, 4, 3, 6]
sage: P.order_ideal([])
[]