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Reference: Combinatorial Commutative Algebra by Miller and Sturmfels.

I have difficulty getting started with Miller and Sturmfels, when I read the first few pages, as I do not know where simplicial complexes come from and would like to find more examples.

In particular, I would like to know how to form subsets of sets as stated in page 4 in Chapter 1 on squarefree monomial ideals.

I would also like to know if Maple can do this.

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There is a topology package, Moise, by R. Andrew Hicks available at – J W Aug 15 '14 at 9:39
up vote 2 down vote accepted

It's perhaps easiest to think of a simplicial complex as a topological rather than a combinatorial object -- in low dimensions at least, it's something you can draw. You start with a set of vertices; connect certain pairs of vertices by edges; fill in certain triples of edges with triangles, also called '2-simplices' in this context; fill in certain quadruples of triangles with tetrahedra or '3-simplices'; and so on$^*$. A simplicial complex is a topological space that can be built by such a process. This definition is equivalent to Miller and Sturmfels': given such a space, the set $\mathbf{n} = \{1,\dotsc,n\}$ they describe is just the set of vertices, and the set $\Delta \subseteq \mathcal{P}(\mathbf{n})$ contains $A \subseteq \mathbf{n}$ iff the simplicial complex has an $|A|$-simplex whose vertices are the elements of $A$.

For example, (the boundary of) a triangle is a simplicial complex with three vertices and three 1-simplices. In terms of the combinatorial definition, it's given by $\Delta = \{\emptyset, \{1\}, \{2\}, \{3\}, \{1,2\}, \{1,3\}, \{2,3\}\}$. If we added $\{1,2,3\}$ to this set, then we'd fill in the boundary to get a solid triangle. Try to do a square and an octahedron if you want practice. The key point is that every subset of an element of $\Delta$ is also an element of $\Delta$ -- in geometric terms, every face of a simplex in the complex also has to be a simplex.

In topology, these sorts of things are useful because they're a rigid, combinatorial way of modeling a large class of spaces. (In fact, if we enlarge the definition slightly to that of a simplicial set, we get a category with the same homotopy theory as that of spaces.) They show up in combinatorics itself a lot as well -- basically any time you have a set of sets that's closed under taking subsets. In this instance, this downward closure is a direct consequence of the definition of 'ideal'. If a monomial $x_{i_1}\dotsm x_{i_n}$ is not in some ideal $I$, then none of its factors are either! Thus if we look at the set of sets $\{i_1,\dotsc,i_n\}$ such that $x_{i_1}\dotsm x_{i_n} \not\in I$, then this is a simplicial complex, and if $I$ is a squarefree monomial ideal, we can entirely recover it from this simplicial complex. This is the content of that first section you're reading. If you're still confused, write down some squarefree monomial ideals and turn them into simplicial complexes!

Hope this helps!

$^*$ To be technical, the simplicial complex isn't just the space; it's the data of where the simplices are as well. A triangle and a square are homeomorphic as spaces, but not isomorphic as simplicial complexes, as they have different numbers of simplices.

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it seems every graph can be set, which kind of numeric data related to these graph – Machine Gun Mar 3 '13 at 11:13
@BeginnerPotato: I can't understand what you're asking me. Could you be a little clearer? – Paul VanKoughnett Mar 4 '13 at 3:16

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