simplicialComplex I
An abstract simplicial complex on a finite set is a family of subsets that is closed under taking subsets. The elements in the abstract simplicial complex are called its faces. The faces having cardinality 1 are its vertices and the maximal faces (order by inclusion) are its facets. Following the combinatorial conventions, every nonempty abstract simplicial complex has the empty set as a face.
In this package, a simplicial complex is represented by its Stanley–Reisner ideal. The vertices are identified with a subset of the variables in a polynomial ring and each face is identified with the product of the corresponding variables. A nonface is any subset of the variables that does not belong to the simplicial complex and each nonface is again identified with a product of variables. The StanleyReisner ideal of a simplicial complex is generated by monomials corresponding to its nonfaces.
The boundary of the 4simplex is a simplicial sphere with 5 vertices, 5 tetrahedral facets, and a minimal nonface that corresponds to the interior of the sphere.






An isomorphic abstract simplicial complex may be constructed in a larger polynomial ring.





The abstract simplicial complex from Example 1.8 of MillerSturmfels' Combinatorial Commutative Algebra consists of a triangle (on vertices $a$, $b$, $c$), two edges (connecting $c$ to $d$ and $b$ to $d$), and an isolated vertex (namely $e$). It has six minimal nonfaces.






There are two "trivial" simplicial complexes: The irrelevant complex has the empty set as a facet whereas the void complex has no facets. Since every abstract simplicial complex in this package is equipped with a chosen polynomial ring, the void complex cannot be constructed by just listing its facets.







Although an abstract simplicial complex can be represented by a Stanley–Reisner ideal in any polynomial ring with a sufficiently large number of variables, some operations in this package do depend of the choice of the polynomial ring (or its coefficient ring). For example, the chain complex of an abstract simplicial complex is typically constructed over the coefficient ring of this polynomial ring, and the dual of a simplicial complex (or monomial ideal) is dependent on the number of variables in the polynomial ideal.



The input can be an Ideal, in which case the constructor verifies that it is a squarefree monomial ideal.
