mjo/clan/trivial_clan.py: new TrivialClan class

Does exactly what you would expect: nothing.

diff --git a/mjo/clan/trivial_clan.py b/mjo/clan/trivial_clan.py
new file mode 100644 (file)
index 0000000..41b5e32
--- /dev/null
+++ b/mjo/clan/trivial_clan.py
@@ -0,0 +1,88 @@
+from mjo.clan.normal_decomposition import NormalDecomposition
+
+class TrivialClan(NormalDecomposition):
+    r"""
+    The trivial (dimension = rank = zero) clan.
+
+    It satisfies the axioms.
+
+    SETUP::
+
+        sage: from mjo.clan.trivial_clan import TrivialClan
+
+    EXAMPLES:
+
+        sage: C = TrivialClan()
+        sage: C.dimension()
+        0
+        sage: C.basis()
+        Finite family {}
+        sage: C.rank()
+        0
+        sage: C.one()
+        0
+
+    TESTS:
+
+    Verifying the axioms (vacuously)::
+
+        sage: C = TrivialClan()
+        sage: e = C.basis()
+        sage: r = C.rank()
+        sage: all( e[i,j,k]*e[i,j,k] == e[i,j,k]
+        ....:      for (i,j,k) in e.keys()
+        ....:      if i == j )
+        True
+        sage: all( C.idempotent(i)*e[j,i,k] == e[j,i,k]/2
+        ....:      for (i,j,k) in e.keys()
+        ....:      if i < j )
+        True
+        sage: all( C.idempotent(i)*e[i,k,z] == e[i,k,z]/2
+        ....:      for (i,k,z) in e.keys()
+        ....:      if k < i)
+        True
+        sage: all( (C.idempotent(i)*e[j,k,l]).is_zero()
+        ....:      for i in range(r)
+        ....:      for (j,k,l) in e.keys()
+        ....:      if k <= j and i not in [j,k] )
+        True
+        sage: all( (e[i,k,l]*C.idempotent(i)).is_zero()
+        ....:      for (i,k,l) in e.keys()
+        ....:      if k < i )
+        True
+        sage: all( (e[j,k,l]*C.idempotent(i)).is_zero()
+        ....:      for i in range(r)
+        ....:      for (j,k,l) in e.keys()
+        ....:      if i not in [j,k] )
+        True
+
+    """
+    from sage.rings.rational_field import QQ
+    def __init__(self, scalar_field=QQ, **kwargs):
+        from sage.modules.free_module import VectorSpace
+        R0 = VectorSpace(scalar_field, [])
+
+        def cp(x,y):
+            return x
+
+        def ip(x,y):
+            return scalar_field.zero()
+
+        super().__init__(R0, cp, ip, **kwargs)
+
+
+    def __repr__(self) -> str:
+        r"""
+        The string representation of this clan.
+
+        SETUP::
+
+            sage: from mjo.clan.trivial_clan import TrivialClan
+
+        EXAMPLES::
+
+            sage: TrivialClan()
+            Trivial clan over Rational Field
+
+        """
+        return f"Trivial clan over {self.base_ring()}"