Preorder


1   package JSci.maths.categories;
2   
3   /**
4   * The Preorder class encapsulates preorders as categories.
5   * @version 1.0
6   * @author Mark Hale
7   */
8   public final class Preorder extends Object   implements Category {
9           private final int N;
10          /**
11          * Constructs a preorder category.
12          */
13          public Preorder(int n) {
14                  N=n;
15          }
16          /**
17          * Returns the identity morphism for an object.
18          * @param a an Integer.
19          */
20          public Category.Morphism identity(Object   a) {
21                  return new Relation((Integer  )a, (Integer  )a);
22          }
23          /**
24          * Returns the cardinality of an object.
25          * @param a an Integer.
26          * @return an Integer.
27          */
28          public Object   cardinality(Object   a) {
29                  return a;
30          }
31          /**
32          * Returns a hom-set.
33          */
34          public Category.HomSet hom(Object   a,Object   b) {
35                  final Integer   i=(Integer  )a;
36                  final Integer   j=(Integer  )b;
37                  if(i.compareTo(j)<=0)
38                          return new RelationSet(i,j);
39                  else
40                          return new RelationSet();
41          }
42          public Object   initial() {
43                  return new Integer  (0);
44          }
45          public Object   terminal() {
46                  return new Integer  (N-1);
47          }
48          /**
49          * Returns the ordinal that this category represents.
50          */
51          public int ordinal() {
52                  return N;
53          }
54          public class RelationSet implements Category.HomSet {
55                  public final Relation morphism;
56                  public RelationSet() {
57                          morphism=null;
58                  }
59                  public RelationSet(Integer   a,Integer   b) {
60                          morphism=new Relation(a,b);
61                  }
62          }
63          public class Relation implements Category.Morphism {
64                  private final Integer   from,to;
65                  public Relation(Integer   a,Integer   b) {
66                          from=a;
67                          to=b;
68                  }
69                  public Object   domain() {
70                          return from;
71                  }
72                  public Object   codomain() {
73                          return to;
74                  }
75                  public Object   map(Object   o) {
76                          return to;
77                  }
78                  public Category.Morphism compose(Category.Morphism m) {
79                          if(m instanceof Relation) {
80                                  Relation r=(Relation)m;
81                                  if(to.equals(r.from))
82                                          return new Relation(from,r.to);
83                                  else
84                                          throw new UndefinedCompositionException();
85                          } else
86                                  throw new IllegalArgumentException  ("Morphism is not a Relation.");
87                  }
88          }
89  }
90  
91
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