gh-37940: Support morphisms in Matroid constructor

    
This follows the merging of #37692 and it enables the (re-)feeding of a
linear matroid's morphism representation into the `Matroid` constructor.

Example:
```
sage: M = matroids.catalog.Fano()
sage: A = M.representation(order=True); A
Generic morphism:
  From: Free module generated by {'a', 'b', 'c', 'd', 'e', 'f', 'g'}
over Finite Field of size 2
  To:   Free module generated by {0, 1, 2} over Finite Field of size 2
sage: Matroid(A)
Binary matroid of rank 3 on 7 elements, type (3, 0)
```
    
URL: #37940
Reported by: gmou3
Reviewer(s): gmou3, Matthias Köppe, Travis Scrimshaw

src/sage/matroids/constructor.py

@@ -188,6 +188,8 @@ def Matroid(groundset=None, data=None, **kwds):
       ``reduced_matrix = A``
       then the matroid is represented by `[I\ \ A]` where `I` is an
       appropriately sized identity matrix.
+    - ``morphism`` -- A morphism representation of the matroid.
+    - ``reduced_morphism`` -- A reduced morphism representation of the matroid.
     - ``rank_function`` -- A function that computes the rank of each subset.
       Can only be provided together with a groundset.
     - ``circuit_closures`` -- Either a list of tuples ``(k, C)`` with ``C``
@@ -533,6 +535,48 @@ def Matroid(groundset=None, data=None, **kwds):
             sage: M.base_ring()
             Integer Ring
 
+        A morphism representation of a :class:`LinearMatroid` can also be used as
+        input::
+
+            sage: M = matroids.catalog.Fano()
+            sage: A = M.representation(order=True); A
+            Generic morphism:
+              From: Free module generated by {'a', 'b', 'c', 'd', 'e', 'f', 'g'} over
+                    Finite Field of size 2
+              To:   Free module generated by {0, 1, 2} over Finite Field of size 2
+            sage: A._unicode_art_matrix()
+              a b c d e f g
+            0⎛1 0 0 0 1 1 1⎞
+            1⎜0 1 0 1 0 1 1⎟
+            2⎝0 0 1 1 1 0 1⎠
+            sage: N = Matroid(A); N
+            Binary matroid of rank 3 on 7 elements, type (3, 0)
+            sage: N.groundset()
+            frozenset({'a', 'b', 'c', 'd', 'e', 'f', 'g'})
+            sage: M == N
+            True
+
+        The keywords ``morphism`` and ``reduced_morphism`` are also available::
+
+            sage: M = matroids.catalog.RelaxedNonFano("abcdefg")
+            sage: A = M.representation(order=True, reduced=True); A
+            Generic morphism:
+              From: Free module generated by {'d', 'e', 'f', 'g'} over
+                    Finite Field in w of size 2^2
+              To:   Free module generated by {'a', 'b', 'c'} over
+                    Finite Field in w of size 2^2
+            sage: A._unicode_art_matrix()
+              d e f g
+            a⎛1 1 0 1⎞
+            b⎜1 0 1 1⎟
+            c⎝0 1 w 1⎠
+            sage: N = Matroid(reduced_morphism=A); N
+            Quaternary matroid of rank 3 on 7 elements
+            sage: N.groundset()
+            frozenset({'a', 'b', 'c', 'd', 'e', 'f', 'g'})
+            sage: M == N
+            True
+
     #.  Rank function:
 
         Any function mapping subsets to integers can be used as input::
@@ -713,8 +757,8 @@ def Matroid(groundset=None, data=None, **kwds):
     if data is None:
         for k in ['bases', 'independent_sets', 'circuits',
                   'nonspanning_circuits', 'flats', 'graph', 'matrix',
-                  'reduced_matrix', 'rank_function', 'revlex',
-                  'circuit_closures', 'matroid']:
+                  'reduced_matrix', 'morphism', 'reduced_morphism',
+                  'rank_function', 'revlex', 'circuit_closures', 'matroid']:
             if k in kwds:
                 data = kwds.pop(k)
                 key = k
@@ -732,8 +776,13 @@ def Matroid(groundset=None, data=None, **kwds):
             Graph = ()
         if isinstance(data, Graph):
             key = 'graph'
-        elif is_Matrix(data):
+        elif is_Matrix(data) or (
+             isinstance(data, tuple) and is_Matrix(data[0])):
             key = 'matrix'
+        elif isinstance(data, sage.modules.with_basis.morphism.ModuleMorphism) or (
+             isinstance(data, tuple) and
+             isinstance(data[0], sage.modules.with_basis.morphism.ModuleMorphism)):
+            key = 'morphism'
         elif isinstance(data, sage.matroids.matroid.Matroid):
             key = 'matroid'
         elif isinstance(data, str):
@@ -856,9 +905,22 @@ def Matroid(groundset=None, data=None, **kwds):
             M = GraphicMatroid(G, groundset=groundset)
 
     # Matrices:
-    elif key in ['matrix', 'reduced_matrix']:
+    elif key in ['matrix', 'reduced_matrix', 'morphism', 'reduced_morphism']:
         A = data
-        is_reduced = (key == 'reduced_matrix')
+        is_reduced = (key == 'reduced_matrix' or key == 'reduced_morphism')
+        if isinstance(data, tuple):
+            A = data[0]
+            if key == 'matrix' or key == 'reduced_matrix':
+                if groundset is None:
+                    groundset = data[1]
+                    if is_reduced:
+                        groundset += data[2]
+        if key == 'morphism' or key == 'reduced_morphism':
+            if groundset is None:
+                groundset = list(A.domain().basis().keys())
+                if is_reduced:
+                    groundset = list(A.codomain().basis().keys()) + groundset
+            A = A.matrix()
 
         # Fix the representation
         if not is_Matrix(A):