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Matroids :: groundSet

groundSet -- (internal) ground set

Synopsis

Description

Returns the internal representation of the ground set.

Important: read the following if you encounter errors when specifying subsets of a matroid (e.g. restriction/deletion/contraction, rank of subset, etc.)

For a matroid M, there are 2 important differences between M.groundSet and the elements of M (given by M_*). First is data types: M.groundSet is a Set, and M_* is a List. Second, M.groundSet always consists of integers from 0 to n-1, where n is the number of elements of M: on the other hand, the elements of M themselves can be arbitrary (e.g. symbols, matrices, edges in a graph, etc.).

Thus, one can think of M.groundSet as the set of indices of the elements in the list M_*: the first element of M has index 0, corresponding to the element 0 in M.groundSet; the second element of M has index 1, etc.

The key point is that all sets associated to the structure of a matroid - bases, circuits, flats, etc. - are subsets of M.groundSet (not M_*). In particular, they are also of class Set (although a collection of them is usually a List), and are also indexed from 0 to n-1. (An exception here is loops and coloops, which are given as a list of indices, rather than single-element sets).

A recommended way to circumvent this distinction between indices and elements is to use $\{0, ..., n-1\}$ as the actual elements of M, in which case an element is equal to its index in M.groundSet. Most methods in this package will accept either a list of elements or a set of indices, and if the elements of M are $\{0, ..., n-1\}$, then functionally there will be no difference between inputting lists or sets.

In summary: lists are used for elements in M, and given as sublists of M_*, while sets are used for indices, and given as subsets of M.groundSet.

i1 : M = matroid({a,b,c,d},{{a,b},{a,c}})

o1 = a matroid of rank 2 on 4 elements

o1 : Matroid
i2 : peek M

o2 = Matroid{bases => {set {0, 1}, set {0, 2}}}
             cache => CacheTable{...2...}
             groundSet => set {0, 1, 2, 3}
             rank => 2
i3 : M.groundSet

o3 = set {0, 1, 2, 3}

o3 : Set
i4 : M_*

o4 = {a, b, c, d}

o4 : List
i5 : bases M

o5 = {set {0, 1}, set {0, 2}}

o5 : List
i6 : (bases M)#0

o6 = set {0, 1}

o6 : Set
i7 : circuits M

o7 = {set {1, 2}, set {3}}

o7 : List
i8 : flats M

o8 = {set {3}, set {0, 3}, set {1, 2, 3}, set {0, 1, 2, 3}}

o8 : List
i9 : loops M

o9 = {3}

o9 : List
i10 : coloops M

o10 = {0}

o10 : List

Note in particular the types of the various outputs above.

The following illustrates how to perform operations with a specified subset of M.groundSet. In the final example, a list of indices is given, which goes against the conventions above, but the elements of the list are treated (correctly) as indices, and if debugLevel is greater than 0, then a warning is printed.

i11 : N1 = M | {a,c,d}

o11 = a matroid of rank 2 on 3 elements

o11 : Matroid
i12 : N2 = M | set{0,2,3}

o12 = a matroid of rank 2 on 3 elements

o12 : Matroid
i13 : N1 == N2

o13 = true
i14 : debugLevel = 1

o14 = 1
i15 : N3 = M | {0,2,3} -- gives a warning, but attempts to treat 0 as an index
Warning: 0 is not a member of {a, b, c, d}
Treating 0 as an index (cf. 'help groundSet' for how to input subsets) ...

o15 = a matroid of rank 2 on 3 elements

o15 : Matroid
i16 : N3 == N2

o16 = true

See also

For the programmer

The object groundSet is a symbol.