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<head><title>Face -- The class of all faces of complexes or co-complexes.</title>
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<div><h1>Face -- The class of all faces of complexes or co-complexes.</h1>
<div class="single"><h2>Description</h2>
<div><p>The class of all faces of <a href="___Complex.html" title="The class of all embedded complexes.">Complex</a>es or <a href="___Co__Complex.html" title="The class of all embedded co-complexes.">CoComplex</a>es.</p>
<p><b>Creating faces:</b></p>
<p>Faces can be created by <a href="_face.html" title="Generate a face.">face</a> by specifying a <a href="../../Macaulay2Doc/html/___List.html" title="the class of all lists -- {...}">List</a> or <a href="../../Macaulay2Doc/html/___Set.html" title="the class of all sets">Set</a> of variables of a <a href="../../Macaulay2Doc/html/___Polynomial__Ring.html" title="the class of all ordered monoid rings">PolynomialRing</a> and some more optional data.</p>
<p>Usually faces are accessed as faces of a <a href="_complex.html" title="Make a complex.">complex</a> C using the <a href="../../Macaulay2Doc/html/___Scripted__Functor.html" title="the class of all scripted functors">ScriptedFunctor</a> C.fc with the subscripts "dimension of the face" and "index of the face".</p>
<p><b>The data stored in a face F:</b></p>
<p><i>F.vert</i>, a list with the vertices of F, which are variables of a <a href="../../Macaulay2Doc/html/___Polynomial__Ring.html" title="the class of all ordered monoid rings">PolynomialRing</a> R. We use a list to prevent Macaualy from reordering sets or monomials.</p>
<p><i>F.ofComplex</i>, a list with the <a href="___Complex.html" title="The class of all embedded complexes.">Complex</a>es and <a href="___Co__Complex.html" title="The class of all embedded co-complexes.">CoComplex</a>es of which F is a face (optional).</p>
<p>it F.indices}, a list with the indices of the face in its complexes. An index is a list dim F, index of F in C.fc<sub>(</sub>dim F).</p>
<p><i>F.dualFace</i>, the dual face of F (optional, is stored for later use when computed by <a href="_dualize.html" title="The dual of a face or complex.">dualize</a>). Note that also the dual face dF of F then has dF.dualFace=F.</p>
<p>This data can also be accessed by the methods listed below.</p>
<p>Note that the lattice data of the vertices is stored in R.grading via <a href="_add__Coker__Grading.html" title="Stores a cokernel grading in a polynomial ring.">addCokerGrading</a>. If no coker grading is present then many functions, like <a href="_simplex.html" title="Simplex in the variables of a polynomial ring.">simplex</a>, <a href="_ideal__To__Complex.html" title="The complex associated to a reduced monomial ideal.">idealToComplex</a>, <a href="_ideal__To__Co__Complex.html" title="The co-complex associated to a reduced monomial ideal.">idealToCoComplex</a> add the standard projective space grading <a href="_rays__P__Pn.html" title="The rays of the standard fan of projective space.">raysPPn</a>.</p>
<p><b>Notes on the implementation:</b></p>
<p>In order to save memory identical faces are not created several times, e.g., if sub<a href="___Complex.html" title="The class of all embedded complexes.">Complex</a>es are created from given ones any face of the subcomplex is identical to a face of the given one.</p>
<p>So far the list F.ofComplex is only used for the complex which created F, but later we will perhaps append also all other complexes F is a face of. As we will append to the list this will be backwards compatible.</p>
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<table class="examples"><tr><td><pre>i1 : R=QQ[x_0..x_4]
o1 = R
o1 : PolynomialRing</pre>
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<tr><td><pre>i2 : addCokerGrading R
o2 = | -1 -1 -1 -1 |
| 1 0 0 0 |
| 0 1 0 0 |
| 0 0 1 0 |
| 0 0 0 1 |
5 4
o2 : Matrix ZZ <--- ZZ</pre>
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<tr><td><pre>i3 : C=simplex R
o3 = 4: x x x x x
0 1 2 3 4
o3 : complex of dim 4 embedded in dim 4 (printing facets)
equidimensional, simplicial, F-vector {1, 5, 10, 10, 5, 1}, Euler = 0</pre>
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<tr><td><pre>i4 : F=C.fc_1_0
o4 = x x
0 1
o4 : face with 2 vertices</pre>
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<tr><td><pre>i5 : F.vert
o5 = {x , x }
0 1
o5 : List</pre>
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<tr><td><pre>i6 : coordinates F
o6 = {{-1, -1, -1, -1}, {1, 0, 0, 0}}
o6 : List</pre>
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<tr><td><pre>i7 : (F.ofComplex)#0==C
o7 = true</pre>
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<tr><td><pre>i8 : bC=boundaryOfPolytope C
o8 = 3: x x x x x x x x x x x x x x x x x x x x
0 1 2 3 0 1 2 4 0 1 3 4 0 2 3 4 1 2 3 4
o8 : complex of dim 3 embedded in dim 4 (printing facets)
equidimensional, simplicial, F-vector {1, 5, 10, 10, 5, 0}, Euler = -1</pre>
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<tr><td><pre>i9 : F==bC.fc_1_0
o9 = true</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="___Complex.html" title="The class of all embedded complexes.">Complex</a> -- The class of all embedded complexes.</span></li>
<li><span><a href="___Co__Complex.html" title="The class of all embedded co-complexes.">CoComplex</a> -- The class of all embedded co-complexes.</span></li>
<li><span><a href="_of__Complex.html" title="The complex of a face.">ofComplex</a> -- The complex of a face.</span></li>
<li><span><a href="_vert.html" title="The vertices of a face or complex.">vert</a> -- The vertices of a face or complex.</span></li>
<li><span><a href="../../Macaulay2Doc/html/_dual.html" title="dual module or map">dual</a> -- dual module or map</span></li>
<li><span><a href="_add__Coker__Grading.html" title="Stores a cokernel grading in a polynomial ring.">addCokerGrading</a> -- Stores a cokernel grading in a polynomial ring.</span></li>
<li><span><a href="_simplex.html" title="Simplex in the variables of a polynomial ring.">simplex</a> -- Simplex in the variables of a polynomial ring.</span></li>
<li><span><a href="_boundary__Of__Polytope.html" title="The boundary of a polytope.">boundaryOfPolytope</a> -- The boundary of a polytope.</span></li>
<li><span><a href="_coordinates.html" title="The coordinates of a face.">coordinates</a> -- The coordinates of a face.</span></li>
</ul>
</div>
<div class="waystouse"><h2>Methods that use a face :</h2>
<ul><li><span>closedStar(Face,Complex), see <span><a href="_closed__Star.html" title="The closed star of a face of a complex.">closedStar</a> -- The closed star of a face of a complex.</span></span></li>
<li><span><a href="_complement_lp__Face_rp.html" title="Compute the complement face of a simplex.">complement(Face)</a> -- Compute the complement face of a simplex.</span></li>
<li><span>coordinates(Face), see <span><a href="_coordinates.html" title="The coordinates of a face.">coordinates</a> -- The coordinates of a face.</span></span></li>
<li><span>coordinates(Face,Complex), see <span><a href="_coordinates.html" title="The coordinates of a face.">coordinates</a> -- The coordinates of a face.</span></span></li>
<li><span>deformationsFace(Face,Complex), see <span><a href="_deformations__Face.html" title="Compute the deformations associated to a face.">deformationsFace</a> -- Compute the deformations associated to a face.</span></span></li>
<li><span>deformationsFace(Face,Complex,Ideal), see <span><a href="_deformations__Face.html" title="Compute the deformations associated to a face.">deformationsFace</a> -- Compute the deformations associated to a face.</span></span></li>
<li><span><a href="_dim_lp__Face_rp.html" title="Compute the dimension of a face.">dim(Face)</a> -- Compute the dimension of a face.</span></li>
<li><span><a href="_dim_lp__Face_cm__Complex_rp.html" title="Compute the dimension of a face.">dim(Face,Complex)</a> -- Compute the dimension of a face.</span></li>
<li><span><a href="_dim_lp__Face_cm__Polynomial__Ring_rp.html" title="Compute the dimension of a face.">dim(Face,PolynomialRing)</a> -- Compute the dimension of a face.</span></li>
<li><span>dualize(Face), see <span><a href="_dualize.html" title="The dual of a face or complex.">dualize</a> -- The dual of a face or complex.</span></span></li>
<li><span><a href="___Face_sp_eq_eq_sp__Face.html" title="Compare two faces.">Face == Face</a> -- Compare two faces.</span></li>
<li><span>faceToMonomial(Face), see <span><a href="_face__To__Monomial.html" title="The monomial of a face.">faceToMonomial</a> -- The monomial of a face.</span></span></li>
<li><span>faceToMonomial(Face,PolynomialRing), see <span><a href="_face__To__Monomial.html" title="The monomial of a face.">faceToMonomial</a> -- The monomial of a face.</span></span></li>
<li><span><a href="_indices_lp__Face_rp.html" title="The indices of a face.">indices(Face)</a> -- The indices of a face.</span></li>
<li><span>intersectFaces(Face,Face), see <span><a href="_intersect__Faces.html" title="The intersection of two faces.">intersectFaces</a> -- The intersection of two faces.</span></span></li>
<li><span>isSubface(Face,Face), see <span><a href="_is__Subface.html" title="Checks whether a face is a subface of another face.">isSubface</a> -- Checks whether a face is a subface of another face.</span></span></li>
<li><span>link(Face,Complex), see <span><a href="_link.html" title="The link of a face of a complex.">link</a> -- The link of a face of a complex.</span></span></li>
<li><span><a href="_net_lp__Face_rp.html" title="Printing faces.">net(Face)</a> -- Printing faces.</span></li>
<li><span>ofComplex(Face), see <span><a href="_of__Complex.html" title="The complex of a face.">ofComplex</a> -- The complex of a face.</span></span></li>
<li><span>simplexDim(Face), see <span><a href="_simplex__Dim.html" title="The dimension of a face inside the vertex simplex.">simplexDim</a> -- The dimension of a face inside the vertex simplex.</span></span></li>
<li><span><a href="_simplex__Ring_lp__Face_rp.html" title="The underlying polynomial ring of a face.">simplexRing(Face)</a> -- The underlying polynomial ring of a face.</span></li>
<li><span>vert(Face), see <span><a href="_vert.html" title="The vertices of a face or complex.">vert</a> -- The vertices of a face or complex.</span></span></li>
</ul>
</div>
<div class="waystouse"><h2>For the programmer</h2>
<p>The object <a href="___Face.html" title="The class of all faces of complexes or co-complexes.">Face</a> is <span>a <a href="../../Macaulay2Doc/html/___Type.html">type</a></span>, with ancestor classes <a href="../../Macaulay2Doc/html/___Mutable__Hash__Table.html" title="the class of all mutable hash tables">MutableHashTable</a> < <a href="../../Macaulay2Doc/html/___Hash__Table.html" title="the class of all hash tables">HashTable</a> < <a href="../../Macaulay2Doc/html/___Thing.html" title="the class of all things">Thing</a>.</p>
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